An introduction to abstract algebra. Provides valuable experience for any further axiomatic study of mathematics. DLC: A
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Table of contents :
A First Course in Abstract Algebra 8th Edition [John B. Fraleigh]
Title Page
Copyright
Contents
Instructor's Preface
Student's Preface
SECTION 0 SETS AND RELATIONS
PART 1 Groups and Subgroups
PART 2 Structure of Groups
PART 3 Homomorphisms and Factor Groups
PART 4 Advanced Group Theory
PART 5 Rings and Fields
PART 6 Constructing Rings and Fields
PART 7 Commutative Algebra
PART 8 Extension Fields
PART 9 Galois Theory
Appendix: Matrix Algebra
Bibliography
Notations
Answers to OddNumbered Exercises Not Asl
A First Course in Abstract Algebra Eighth Edition
John B. Fraleigh University of Rhode Island Neal Brand University of North Texas Historical Notes by Victor Katz University of District of Columbia
@Pearson
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Library of Congress CataloginginPublication Data Names: Fraleigh, John B., author. I Katz, Victor J., writer of added commentary. Title: A first course in abstract algebra I John B. Fraleigh ; historical notes by Victor Katz. Description: Eighth edition. I [Hoboken, New Jersey]: Pearson, (202 11 1 Series: World student series I Includes bibliographical references and index. Identifiers: LCCN 2019038536 I ISBN 9780135758 168 (paperback) I ISBN 978032 1390363 (ebook) Subjects: LCSH: Algebra, Abstract. Classification: LCC QA162 .F7 2020 I DOC 5 12/.02. We have shown a string winding its way through this array. Imagine the fractions to be glued to this string. Taking the beginning of the string and pulling to the left in the direction of the arrow, the string straightens I , 1 , ~, · · · . Thus out and all elements of IQ> appear on it in an infinite row as 0, llQ>I = ~o also. If the set S = (x E lR I 0 < x < 1} has cardinality ~ 0 , all its elements could be listed as unending decimals in a column extending infinitely downward, perhaps as
1, 1,
0.3659663426 .. . 0.7103958453 .. . 0.0358493553 .. . 0.99684522 14 .. .
We now argue that any such array must omit some number in S. Surely S contains a number r having as its nth digit after the decimal point a number different from 0, from 9, and from the nth digit of the nth number in this list. For example, r might start .5637· · · . The 5 rather than 3 after the decimal point shows r cannot be the first number in S listed in the array shown. The 6 rather than 1 in the second digit shows r cannot be the second number listed, and so on. Because we could make this argument with any list, we see that S has too many elements to be paired with those in z+. Exercise 15 indicates that lR has the same number of elements as S. We j ust denote the cardinality of lR by llRI. Exercise 19 indicates that there are infinitely many different cardinal numbers even greater than llRI.
Partitions and Equivalence Relations Sets are disjoint if no two of them share a common element. In Example 0. 17 we break up the integers into subsets. Eventually we will see how to define an algebraic structure on these subsets of Z . That is, we will be able to "add" two of these subsets to get another subset. We will find that breaking a set into subsets is a valuable tool in a number of settings, so we conclude this section with a brief study of partitions of sets. 0.16 Definition
A partition of a set S is a collection of nonempty subsets of S such that every element of S is in exactly one of the subsets. The subsets are the cells of the partition. •
6
Section 0
Sets and Relations
When discussing a partition of a set S, we denote by x the cell containing the element x of S.
0.17 Example
Splitting Z into the subset of even integers and the subset of odd integers, we obtain a partition of Z into the two cells listed below.
0 = {... , 8, 6, 4, 2, 0, 2, 4, ... } T = {... , 7, 5, 3, 1, 1, 3, 5, . .. l We can think of 0 as being the integers that are divisible by 2 and T as the integers that when divided by 2 yield a remainder of 1. This idea can be used for positive integers other than 2. For example, we can partition Z into three cells:
= {x E Z Ix is a multiple of 3), T = {x E Z I the remainder of x divided by 3 is 1), 2 = {x E Z I the remainder of x divided by 3 is 2).
0
and
Note that when dividing a negative number by 3, we still obtain a nonnegative remainder. For example, 5: 3 is 2 with remainder 1, which says that 5 = T. __ generalizing, for each n E z+, we obtain a partition of Z consisting of n cells, 0, 1, 2, ... , n  1. For each 0 ::::: r ::::: n  1, an integer x is in the cell r exactly when the remainder of x: n is r. These cells are the residue classes modulo n in Z and n is called the modulus. We define the set Z/nZ as the set containing the cells in this partition. So, for example, Z/3Z = {0, T, 2). As we can see, Z/nZ = (0, T, 2, ... , n  1) has exactly n elements. •
..n
on S in a natural way: namely, for Each partition of a set S yields a relation y if and only if x and y are in the same cell of the partition. In set notation, we would write x ..96 y as (x,y) E ..96 (see Definition 0.7). A bit of thought shows that this relation ..n on S satisfies the three properties of an equivalence relation in the following definition.
x, y E S, let x
0.18 Definition
..n
An equivalence relation ..96 on a set S is one that satisfies these three properties for all x,y,z ES.
1. (Reflexive) x
..n x.
2.
(Symmetric) If x ..98 y , then y ..98 x.
3.
(Transitive) If x ..96 y and y ..96 z then x
•
..n z.
..n
To illustrate why the relation corresponding to a partition of S satisfies the symmetric condition in the definition, we need only observe that if y is in the same cell as x (that is, if x ..96 y), then x is in the same cell as y (that is, y x). We leave the similar observations to verify the reflexive and transitive properties to Exercise 28.
..n
0.19 Example
For any nonempty set S, the equality relation = defined by the subset {(x, x) Ix E S} of S x S is an equivalence relation. •
0.20 Example
(Congruence Modulo n) Let n E z+. The equivalence relation on Z corresponding to the partition of Z into residue classes modulo n, discussed in Example 0. 17, is congruence modulo n. It is sometimes denoted by =,,. Rather than write a = ,,b, we usually write a= b (mod n), read, "a is congruent to b modulo n." For example, we have 15 27 (mod 4) because both 15 and 27 have remainder 3 when divided by 4. •
=
7
Exercises
0.21 Example
..n m if and only if nm '.:: 0, and let us
Let a relation 5B on the set Z be defined by n determine whether is an equivalence relation.
..n
Reflexive Symmetric Transitive
a 5B a, because a 2 ::=: 0 for all a
E
Z.
..n a. ..n b and b ..n c, then ab ::=: 0 and be ::=: 0. Thus ab2 c = acb2 ::=: 0.
If a 5B b, then ab '.:: 0, so ba '.:: 0 and b If a
If we knew b 2 > 0, we could deduce ac ::=: 0 whence a 5B c. We have to examine the case b = 0 separately. A moment of thought shows that 3 5B 0 and 0 5, but we do not have 3 5B 5. Thus the relation is not transitive, and hence is not an .A equivalence relation.
..n
..n
We observed above that a partition yields a natural equivalence relation. We now show that an equivalence relation on a set yields a natural partition of the set. The theorem that follows states both results for reference. 0.22 Theorem
(Equivalence Relations and Partitions) Let S be a nonempty set and let equivalence relation on S. Then ~ yields a partition of S, where
~
be an
a= {x Es Ix~ a}. Also, each partition of S gives rise to an equivalence relation ~ on S where a only if a and b are in the same cell of the partition. Proof
a
~
b if and
We must show that the different cells = {x E S I x ~ a} for a E S do give a partition of S, so that every element of S is in some cell and so that if a E b, then = b. Let a E S. Then a Ea by the reflexive _condition(!), so a is in at lea_!t one cell. Suppose now that a E b. We need to show that ii = b as sets; this will show that a cannot be in more than one cell. There is a standard way to show that two sets are the same:
a
Show that each set is a subset of the other. We show that a ~ b. Let x E ~Then x ~ q. But a E b, so a ~ b._Then, by the tr~nsitive condition (3), x_ ~ b, so x E b. Thus ii ~ b. Now we show that b ~ ii. Let y E b. Then y ~b. But a E b, so a ~ band, by symmetry (2), b ~a. Then by transitivity (3), y ~a, so y E ii. Hence b ~ a also, so b = a and our proof is complete. +

Each cell in the partition arising from an equivalence relation is an equivalence class.
• EXERCISES 0 In Exercises 1 through 4, describe the set by listing its elements.
1. {xE 1Rl x2 =3}
2. {mE Z lm2 +m=6}
3. {m E Z Imn = 60 for some n E Z }
4. {x E Z
Ix2 
lOx + 16 :'.';'. 0)
In Exercises 5 through I 0, decide whether the object described is indeed a set (is well defined). Give an alternate description of each set. 5. {n 7. {n
E E
z+ I n is a large number} Z 139
0, then T/ : D* + Z defined by TJ(a) = v(a) + s for nonzero a E D is a Euclidean norm on D. As usual , D* is the set of nonzero elements of D . b. Show that fort E z+, A. : D* + Z given by A.(a) = t · v(a) for nonzero a E D is a Euclidean norm on D. c. Show that there exists a Euclidean normµ, on D such that µ,(I) = I and µ,(a) > I 00 for all nonzero nonu nits a E D. 20. Let D be a UFD. An element c in D is a least common multiple (abbreviated lcm) of two elements a and b in D if a I c, b I c and if c divides every element of D that is divisible by both a and b. Show that every two no nzero elements a and b of a Euclidean domain D have an !cm in D. [Hint: Show that all common multiples, in the obvious sense, of both a and b form an ideal of D.] 21. Use the last statement in Theorem 35.9 to show that two nonzero elements r, s E Z generate the group (Z, +) if and only if rand s, viewed as integers in the domain Z, are relatively prime, that is, have a gcd of I . 22. Using the last statement in Theorem 35.9, show that for nonzero a, b, n has a solution in Z if a and n are relatively prime.
E
Z , the congruence ax
= b (mod n)
=
23. Generalize Exercise 22 by showing that for nonzero a, b, n E Z, the congruence ax b (mod n) has a solution in Z if and only if the positive gcd of a and n in Z divides b. Interpret this result in the ring Z 11 • 24. Following the idea of Exercises 6 and 23, outline a constructive method for finding a solution in Z of the congruence ax = b (mod n) for nonzero a, b, n E Z, if the congruence does have a solution. Use this method to find a solution of the congruence 22x 18 (mod 42).
=
292
Part VII
SECTION
36
Commutative Algebra
NUMBER THEORY In this section we will show how the ideas in Section 35 can be used to derive some interesting results in number theory. We usually think of number theory as a study of properties of the integers, but Gauss expanded the study of numbers to include what is now called the Gaussian integers. The Gaussian integers form a subring of the complex numbers that, like the integers, form a Euclidean domain, but not a field. After studying the Gaussian integers, we will prove that for any prime number p E z+ that is equivalent to 1 modulo 4, p can be written as the sum of two squares.
Gaussian Integers 36.1 Definition
A Gaussian integer is a complex number a + bi, where a, b a= a+ bi, the norm N(a) of a is a2
E
Z. For a Gaussian integer
+ b2 .
•
Although we defined N(a) for a Gaussian integer a , we can also think of N as defined on any complex number using the same formu la N(a + bi) = a 2 + b 2 . This norm can also be written as N(a) = lal 2 . We shall let Z[i] be the set of all Gaussian integers. The following lemma gives some basic properties of the norm function N on Z[i] and leads to a demonstration that the function v defined by v(a) = N(a) for nonzero a E Z [i] is a Euclidean norm on Z [i]. Note that the Gaussian integers include all the rational integers, that is, all the elements of Z.
• HISTORICAL NOTE
I
n his Disquisitiones Arithmeticae, Gauss studied in detail the theory of quadratic residues, that is, the theory of solutions to the congruence x2 p (mod q) and proved the famous quadratic reciprocity theorem showing the relationship between the solutions of the congruences x2 p (mod q) and x 2 q (mod p) where p and q are primes. In attempting to generalize his results to theories of quartic residues, however, Gauss realized that it was much more natural to consider the Gaussian integers rather than the ordinary integers. Gauss's investigations of the Gaussian integers are contained in a long paper published in 1832 in which he proved various analogies between them and the ordinary integers. For example, after noting that there are four units (invertible elements)
=
=
=
36.2 Lemma
among the Gaussian integers, namely 1, 1 , i, and i, and defining the norm as in Definition 36.1 , he generalized the notion of a prime integer by defining a prime Gaussian integer to be one that cannot be expressed as the product of two other integers, neither of them units. He was then able to determine which Gaussian integers are prime: A Gaussian integer that is not real is prime if and only if its norm is a real prime, which can only be 2 or of the form 4n + 1. The real prime 2 = (1 + i)(l  i) and real primes congruent to 1 modulo 4 like 13 = (2 + 3i)(2  3i) factor as the product of two Gaussian primes. Real primes of the form 4n + 3 like 7 and 11 are still prime in the domain of Gaussian integers. See Exercise 10.
In Z [i], the following properties of the norm function N hold for all a, f3 E Z[i]: 1. N(a) :'.': 0.
Proof
= 0 if and only if a = 0. = N(a)N(f3).
2.
N(a)
3.
N(af3)
= a 1 + a 2 i and f3 = b 1 + b 2 i, these results are all straightforward computations. We leave the proof of these properties as an exercise (see Exercise 11). +
If we let a
Section 36
Number Theory
293
The proof of Lemma 36.2 does not depend on the complex numbers a and f3 being Gaussian integers. In fact the three properties listed in the lemma are true for all complex numbers. 36.3 Lemma Proof
Z [i] is an integral domain. It is obvious that Z [i] is a commutative ring with unity. We show that there are no divisors of 0. Let a , f3 E Z [i]. Using Lemma 36.2, if af3 = 0 then
= N(af3) = N(O) = 0. Thus af3 = 0 implies that N(a) = 0 or N(f3) = 0. By Lemma 36.2 again, this implies that either a = 0 or f3 = 0. Thus Z [i] has no divisors of 0, so Z [i] is an integral + domain. N(a)N(f3)
Of course, since Z[i] is a subring of C, where C is the field of complex numbers, it is really obvious that Z [i] has no 0 divisors. We gave the argument of Lemma 36.3 to illustrate the use of the multiplicative property 3 of the norm function N and to avoid going outside of Z [i] in our argument. However, in the proof of Theorem 36.4, we will use property 3 for complex numbers that are not Gaussian integers and therefore we will stay outside the Gaussian integers. 36.4 Theorem
The function v given by v(a) = N(a) for nonzero a E Z [i] is a Euclidean norm on Z [i]. Thus Z [i] is a Euclidean domain .
Proof
Note that for f3 = b 1 + b2 i :f. 0, N(b 1 + b2 i) = b 12 + b2 2 , so N(f3) '.'.': 1. Then for all a, f3 :f. 0 in Z [i], N(a) :::: N(a)N(f3) = N(af3). This proves Condition 2 for a Euclidean norm in Definition 35.1. It remains to prove the division algorithm, Condition 1, for N. Let a, f3 E Z [i], with a = a 1 + a 2 i and f3 = b 1 + b2 i, where f3 :f. 0. We must find u and p in Z [i] such that a= {Ju+ p, where either p = 0 or N(p) < N(f3) = b 12 + b22 . Let a /{3 = r + si for r, s E IQ. Let q 1 and q 2 be integers in Z as close as possible to the rational numbers rand s, respectively. Let u = q 1 + q2 i and p =a  {Ju. If p = 0, we are done. Otherwise, by construction of u, we see that lr  q 1 I :::: ~ and Is  qzl ::: ~Therefore
N(~ 
u) =
N((r + si)  (q 1 + qzi))
(D + G) 2
=N((r  q i )+(sq2 )i)::::
2
2
Thus we obtain N(p)
= N(a
 {Ju)=
N(/J(~

u))
so we do indeed have N(p) < N(f3) as desired. 36.5 Example
= N(f3)N(~ 
u) :S
N(f3)~,
•
We can now apply all our results of Section 35 to Z [i]. In particular, since N(l) = 1, the units of Z [i] are exactly the a = a 1 + a 2 i with N(a) = a 12 + a 22 = 1. From the fact that a 1 and a 2 are integers, it follows that the only possibilities are a 1 = ±1 with a 2 = 0, or a 1 = 0 with a 2 = ±1. Thus the units of Z [i] are ±1 and ±i. One can also use the Euclidean Algorithm to compute a gcd of two nonzero elements. We leave such computations to the exercises. Finally, note that while 5 is an irreducible in Z, 5 is no longer an irreducible in Z [i], for 5 = (1 + 2i)(l  2i), and neither 1 + 2i nor .& 1  2i is a unit.
294
Part VII
Commutative Algebra
Multiplicative Norms Let us point out again that for an integral domain D , the arithmetic concepts of irreducibles and units are not affected in any way by a norm that may be defined on the domain. However, as the preceding section and our work thus far in this section show, a suitably defined norm may be of help in determining the arithmetic structure of D. This is strikingly illustrated in algebraic number theory, where for a domain of algebraic integers we consider many different norms of the domain, each doing its part in helping to uncover the arithmetic structure of the domain. In a domain of algebraic integers, we have essentially one norm for each irreducible (up to associates), and each such norm gives information concerning the behavior in the integral domain of the irreducible to which it corresponds. This is an example of the importance of studying properties of elements in an algebraic structure by means of mappings associated with them. Let us study integral domains that have a multiplicative norm satisfying Properties 2 and 3 of Non Z[i] given in Lemma 36.2. 36.6 Definition
36.7 Theorem
Proof
Let D be an integral domain. A multiplicative norm N on D is a function mapping D into the integers Z such that the following conditions are satisfied:
= 0 if and only if a = 0. = N(et)N(f3) for all et , f3
1.
N(a)
2.
N(etf3)
•
ED.
If Dis an integral domain with a multiplicative norm N , then N(I ) = 1 and IN(u)I = 1 for every unit u in D. If, furthermore, every a such that IN(a)I = I is a unit in D , then an element n in D, with IN(rr)I = p for a prime p E Z, is an irreducible of D.
Let D be an integral domain with a multiplicative norm N. Then N(l) shows that N(I)
=
= N((l)(l)) = N(l)N(I)
1. Also, if u is a unit in D, then 1 = N( l)
= N(uu 1) = N(u)N(u 1).
Since N(u) is an integer, this implies that IN(u) I = 1. Now suppose that the units of D are exactly the elements of norm ± 1. Let n be such that IN(rr) I = p, where p is a prime in Z. Then if n = af3, we have
= IN(rr) I = IN(a)N(f3) 1= IN(a)l lN(fJ) I, so either IN(a)I = 1 or IN(f3)1 = 1. By assumption, this means that either a
E
D
p
unit of D . Thus n is an irreducible of D . 36.8 Example
or f3 is a
+
On Z[i], the function N defined by N(a + bi) = a 2 + b 2 gives a multiplicative norm in the sense of our definition. We saw that the function v given by v(a) = N(a) for nonzero a E Z[i] is a Euclidean norm on Z [i], so the units are precisely the elements a of Z [i] with N (a) = N(I) = 1. Thus the second part of Theorem 36.7 applies in Z[i]. We saw in Example 36.5 that 5 is not an irreducible in Z [i], for 5 = (I + 2i)(l  2i). Since N(l + 2i) = N(l  2i) = 12 + 22 = 5 and 5 is a prime in Z, we see from Theorem 36.7 that 1 + 2i and 1  2i are both irreducibles in Z [i]. .._ As an application of mutiplicative norms, we shall now give another example of an integral domain that is not a UFD. We saw one example in Example 34. 17. The following is the standard illustration.
36.9 Example
LetZ[H ] ={a+ ib.../51 a,b E Z). As a subset of the complex numbers closed under addition, subtraction, and multiplication, and containing 0 and 1, Z [H J is an integral
Section 36
domain. Define N on Z [H
Number Theory
295
J by N(a
+ b.../=5) =
a2
+ 5b2 .
(Here H = i,JS.) Clearly, N(a ) = 0 if and only if a =a+ b H = 0. That N(af3) = N(a)N(f3) is a straightforward computation that we leave to the exercises (see Exercise 12). Let us find all candidates for units in Z [H J by finding all elements a in Z[HJ with N(a) = 1. If a= a+ b H, and N(a) = 1, we must have a2 + 5b2 = 1 for integers a and b. This is possible only if b = 0 and a= ±1. Hence ±1 are the only candidates for units. Since ± 1 are units, they are then precisely the units in
Z [HJ. Now in Z[HJ, we have 21 = (3)(7) and also
21 = (1 + 2.../=5)(1  2.../=5). If we can show that 3, 7, 1 + 2H, and 1  2 H are all irreducibles in Z[H J, we will then know that Z [H J cannot be a UFD, since neither 3 nor 7 is ±(1 + 2H). Suppose that 3 = af3. Then
= N(3) = N(a)N(f3 ) shows that we must have N(a) = 1, 3, or 9. If N (a) = 1, then a is a unit. If a = a + b H, then N(a) = a 2 + 5b2 , and for no choice of integers a and bis N(a) = 3. If 9
N(a ) = 9, then N(f3) = 1, so f3 is a unit. Thus from 3 = af3, we can conclude that either a or f3 is a unit. Therefore, 3 is an irreducible in Z[HJ. A similar argument shows that 7 is also an irreducible in Z[HJ. If 1 + 2 H = y 8, we have
21 = N (I + 2.../=5) = N( y)N(8). so N( y ) = I , 3, 7, or 21. We have seen that there is no element of Z [H J of norm 3 or 7. Thus either N(y) = 1, and y is a unit, or N(y) = 21, so N (8) = 1, and 8 is a unit. Therefore, 1+2H is an irreducible in Z [HJ. A parallel argument shows that 1  2 H is also an irreducible in Z [HJ. In summary, we have shown that
Z [.../=SJ ={a+ ib.JS i a, b
E
Z}
is an integral domain but not a UFD. In particular, there are two different factorizations
21 = 3. 7 = (1 + 2.../=5)(1  2.../=5) of 21 into irreducibles. These irreducibles cannot be primes, for the property of a prime enables us to prove uniqueness of factorization (see the proof of Theorem 34.18). .A We conclude with a classical application, determining which primes p in Z are equal to a sum of squares of two integers in Z. For example, 2 = 12 + 12 , 5 = 12 + 22 , and 13 = 22 + 32 are sums of squares. Since we have now answered thi s question for the only even prime number, 2, we can restrict ourselves to odd primes. 36.10 Theorem
Proof
(Fermat's p = a 2 + b 2 Theorem) Let p be an odd prime in Z. Then p integers a and bin Z if and only if p 1 (mod 4).
=
= a2 + b2 for
First, suppose that p = a 2 + b2 . Now a and b cannot both be even or both be odd since p is an odd number. If a= 2r and b = 2s + 1, then a2 + b2 = 4? + 4(s 2 + s) + 1, so p 1 (mod 4). This takes care of one direction for this "if and only if" theorem. 1 (mod 4). Now the multiplicative For the other direction, we assume that p group of nonzero elements of the finite field Zp is cyclic, and has order p  1. Since
=
=
296
Part VII
Commutative Algebra
4 is a divisor of p  1, we see that Zp contains an element n of multiplicative order 4. It follows that n2 has multiplicative order 2, so n 2 = 1 in z,,. Thus in Z , we have n2 1 (mod p), sop divides n 2 + 1 in Z . Viewing p and n 2 + 1 in Z [i], we see that p divides n 2 + 1 = (n + i)(n  i). Suppose that p is irreducible in Z [i] ; then p would have to divide n + i or n  i. If p divides n + i, then n + i = p(a + bi) for some a, b E Z . Equating coefficients of i, we obtain 1 = pb, which is impossible. Similarly, p divides n  i would lead to an impossible equation 1 = pb. Thus our assumption that p is irreducible in Z [i] must be false. Since p is not irreducible in Z [i], we have p = (a+ bi)(c +di) where neither a+ bi nor c +di is a unit. Taking norms, we have p 2 = (a 2 + b 2 )(c2 + d2) where neither a2 + b2 = 1 nor c 2 + d 2 = 1. Consequently, we have p = a2 + b2 , which completes our proof. [Since a 2 + b2 = (a + bi)(a  bi), we see that this is the factorization of p , that is, c +di= a  bi.] +
=
Exercise 10 asks you to determine which primes p in 'll, remain irreducible in Z [i].
• EXERCISES 36 Computations In Exercises 1 through 4, factor the Gaussian integer into a product of irreducibles in Z [i]. [Hi nt: Since an irreducible factor of a E Z [i] must have norm > l and dividing N(a) , there are only a finite number of Gaussian integers a+ bi to consider as possible irreducible factors of a given a. Divide a by each of them in IC, and see for which ones the quotient is again in Z[i].]
1. 5
2. 7
4. 6 7i
3. 4+3i
5. Show that 6 does not factor uniquely (up to associates) into irreducibles in Z[.J="S]. Exhibit two different factorizations.
6. Consider a = 7 + 2i and
fJ = 3  4i in Z [i]. Find CJ and pin Z[i] such that a=
{JCJ
+p
with
N(p) < N({J ).
[Hint: Use the construction in the proof of Theorem 36.4.
7. Use a Euclidean algorithm in Z [i] to find a gcd of 8 + 6i and 5  15i in Z [i]. [Hint: Use the construction in the proof of Theorem 36.4.]
Concepts 8. Determine whether each of the following is true or false.
a. Z[i] is a PID. b. Z[i] is a Euclidean domain. c. Every integer in Z is a Gaussian integer. d. Every complex number is a Gaussian integer. e. A Euclidean algorithm holds in Z [i]. f. A multiplicative norm on an integral domain is sometimes an aid in finding irreducibles of the domain. g. If N is a multiplicative norm on an integral domain D, then IN(u)I = 1 for every unit u of D. h. If Fis a field, then the function N defined by N(f(x)) = (degree ofj(x)) is a multiplicative norm on F[x]. i. If Fis a field, then the function defined by N(f(x)) = 2 1 for fJ ED. Show that :rr is an irreducible of D.
Section 37
Algebraic Geometry
297
10. a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i]. b. Show that an odd prime pin Z is irreducible in Z[i] if and only if p = 3 (mod 4). (Use Theorem 36.10.) 11. Prove Lemma 36.2. 12. Prove that N of Example 36.9 is multiplicative, that is, that N(a{J) = N(a)N({J) for a , fJ
E
Z (HJ.
13. Let D be an integral domain with a multiplicative norm N such that IN(a)I = l for a E D if and only if a is a unit of D . Show that every nonzero nonunit of D has a factorization into irreducibles in D.
14. Use a Euclidean algorithm in Z [i] to find a gcd of 16 + 7i and 10  Si in Z[i] . [Hint: Use the construction in the proof of Theorem 36.4.] 15. Let (a) be a nonzero principal ideal in Z [i].
a. Show that Z[i]/ (a) is a finite ring. [Hint: Use the division algorithm.] b. Show that if a is an irreducible of Z(i], then Z [i]/ (a) is a field. c. Referring to part (b), find the order and characteristic of each of the following fields. i. Z (i]/(3)
ii. Z(i] /( l
+ i)
+ 2i) Let Z (Fnl = {a +
iii. Z (i]/( l
16. Let n E z+ be square free, that is, not divisible by the square of any prime integer. ib,./fi. I a, b E Z }.
a. Show thatthe norm N, defined by N(a) = a2 + nb2 fora =a+ ib,./fi., is a multiplicative norm on Z[J=ll]. b. Show thatN(a) = 1 fora E Z [J=ll] ifand only if a is a unit of Z[J=ll]. c. Show that every nonzero a E Z [J=ll] that is not a unit has a factorization into irreducibles in Z[J=ll]. [Hint: Use part (b).] 17. Repeat Exercise 16 for Z (,.fa] = {a + b,./fi. Ia, b E Z } for square free n > 1, with N defined by N(a) = a 2  nb2 for a =a+ b,./fi. in Z[,./fi.]. For part b show IN(a) I = l. 18. Show by a construction analogous to that given in the proof of Theorem 36.4 that the division algorithm holds in the integral domain Z[Nl for v(a) = N(a) for nonzero a in this domain (see Exercise 16). (Thus this
domain is Euclidean. See Hardy and Wright (29] for a discussion of which domains Z [,./fi.] and Z [J=ll] are Euclidean.)
SECTION
37 t ALGEBRAIC GEOMETRY This section gives a brief introduction to algebraic geometry. Algebraic geometry is the study of the common zeros of a finite collection of polynomials. For example, the zeros of the set of polynomials {.x2 + y2  25, (x  6)2 + y2  25} consist of just two points in IR.2 , (3, 4) and (3, 4). In Section 38 we will develop a very useful algorithm that reduces a finite set of polynomials to a simpler set of polynomials whose zeros are identical to the zeros of the original set. In the example {.x2 + y 2  25, (x  6)2 + y 2  25), the algorithm yields {x  3,y2  16} making it much easier to see the two zeros.
Algebraic Varieties and Ideals Let F be a field. Recall that F[x 1, x2, · · · , x 11 ] is the ring of polynomials in n in determinants x 1, x 2, · · · , x 11 with coefficients in F. We let F" be the Cartesian product F x F x · · · x F for n factors. For ease in writing, we denote an element (a 1, a2 , · · · , a 11 ) of F" by a, in bold type. Using similar economy, we let F[x] = F[xi,x2, · · · ,x11 ]. For each a E F 11 , we have an evaluation homomorphism x 1. For any total ordering of all the power products, A Grobner basis can be used to determine if a graph can be colored with n colors starting with a basis consisting of polynomials each of degree at most n. A Grobner basis can be used to determine if a graph can be colored with n colors starting with a basis consisting of r + s polynomials where r is the number of vertices in the graph and sis the number of edges in the graph. I have computed Grobner bases before I knew what they were. Any ideal in F[x] has a unique Grobner basis. A basis for an ideal I in F[x 1,xz, ... ,x11 ] is a Grobner basis if and only if each polynomial in the basis cannot be reduced further using the division algorithm.
xf
28. Let IR[x, y] be ordered by Jex. Give an example to show that P; < Pj does not imply that P; divides Pj. 29. What other orders of the indeterminate a, c, x, y, d1, dz would you expect the equation of an ellipse to result from computing a Grobner basis for the ideal in Example 38.7? 30. Use a Grobner basis to derive the formula for a hyperbola in standard position. Recall that a hyperbola in standard position is the set of all points in the plane whose difference in distances from (c, 0) and ( c, 0) is ±2a. You may use a computer to compute the Grobner basis. 31. Use a Grobner basis to show that the graph with vertex set {x 1,x2,x3,x4,x5 ) and edge set {{x 1,x2 }, {x2,x3}, {x3,x4}, {x1 , x3}, {x 1,x5 ), {x5,x4)} cannot be colored with three colors, but it can be colored with four colors. You may use a computer to compute the Grobner basis.
Theory 32. Show that {.xy, y 2  y) is a Grobner basis for (.xy, y 2  y), as asserted after Example 38.1. 33. Show that {4yp+x2,dy p) is a Grobner basis for the ideal (4yp+x 2,dyp) as asserted in Example 38.6. 34. Prove Theorem 38.8. [Hint: Think about coloring a graph with the nth roots of unity.]
PART
VIII
SECTION
39
Extension Fields
Section 39
Introduction to Extension Fields
Section 40
Algebraic Extensions
Section 41
t Geometric Constructions
Section 42
Finite Fields
INTRODUCTION TO EXTENSION FIELDS
Our Basic Goal Achieved We are now in a position to achieve our basic goal, which, loosely stated, is to show that every nonconstant polynomial has a zero. This will be stated more precisely and proved in Theorem 39.3. We first introduce some new terminology for some old ideas.
39.1 Definition
•
A field E is an extension field of a field F if F :S E. IC
I
lR
I
Q
F(x, y)
/ ~ ~/
F(x)
F(y)
F
39.2 Figure
Thus R is an extension field of Q , and C is an extension field of both R and Q . As in the study of groups, it will often be convenient to use subfield diagrams to picture extension fields, the larger field being on top. We illustrate this in Fig. 39.2. (Recall that F(x) is the field of quotients constructed from F [x].) A configuration where there is just one single column of fields, as at the lefthand side of Fig. 39.2, is often referred to, without any precise definition, as a tower of fields. Now for our basic goal! This great and important result follows quickly and elegantly from the techniques we now have at our disposal.
t Section 4 1 is not required for the remainder of the text.
311
312
Part VIII
Extension Fields
39.3 Theorem
(Kronecker's Theorem) (Basic Goal) Let F be a field and letf(x) be a nonconstant polynomial in F[x]. Then there exists an extension field E of F and an ex E E such that f(ex) = 0.
Proof
By Theorem 28.21,f(x) has a factorization in F[x] into polynomials that are irreducible over F. Let p(x) be an irreducible polynomial in such a factorization. It is clearly sufficient to find an extension field E of F containing an element ex such that p(ex) = 0. By Theorem 31.25, (p(x)) is a maximal ideal in F[x], so F[x]/(p(x)) is a field. We claim that F can be identified with a subfield of F[x]/ (p(x)) in a natural way by use of the map if! : F ~ F[x] /(p(x)) given by if!(a) =a+ (p(x)) for a E F. This map is onetoone, for if if!(a) = if!(b), that is, if a+ (p(x)) = b + (p(x)) for some a, b E F, then (a  b) E (p(x)), so a  b must be a multiple of the polynomial p(x), which is of degree :'.': 1. Now a, b E F implies that a  b is in F. Thus we must have a  b = 0, so a = b. We defined addition and multiplication in F[x]/ (p(x)) by choosing any representatives, so we may choose a E (a+ (p(x)) ). Thus if! is a homomorphism that maps F onetoone onto a subfield of F[x] /(p(x)). We identify F with {a+ (p(x)) I a E F} by means of this map if!. Thus we shall view E = F[x]/ (p(x)) as an extension field of F. We have now manufactured our desired extension field E of F. It remains for us to show that E contains a zero of p(x).
• HISTORICAL NOTE
L
eopold Kronecker is known for his insistence on constructibility of mathematical objects. As he noted, "God made the integers; all else is the work of man." Thus, he wanted to be able to construct new "domains of rationality" (fields) by using only the existence of integers and indeterminates. He did not believe in starting with the real or complex numbers, because as far as he was concerned, those fields could not be determined in a constructive way. Hence in an 1881 paper, Kronecker created an extension field by simply adjoining to a given field a root ex of an irreducible nth degree polynomial p(x); that is, his new field consisted of expressions
rational in the original field elements and his new root ex with the condition that p(ex) = 0. The proof of the theorem presented in the text (Theorem 39.3) dates from the twentieth century. Kronecker completed his dissertation in 1845 at the University of Berlin. For many years thereafter, he managed the family business, ultimately becoming financially independent. He then returned to Berlin, where he was elected to the Academy of Sciences and thus permitted to lecture at the university. On the retirement of Kummer, he became a professor at Berlin, and with Karl Weierstrass (1815 1897) directed the influential mathematics seminar.
Let us set ex = x + (p(x)) ,
so ex EE. Consider the evaluation homomorphism a : F[x] ~ E, given by Theorem 27.4. If p(x) = ao + a 1x + · · · + a11.x'1 , where a; E F, then we have a(p(x))
= ao +a, (x + (p(x))) + · · · + a
11
(x + (p(x)) )
11
in E = F[x] / (p(x)). But we can compute in F[x] / (p(x)) by choosing representatives, and x is a representative of the coset ex = x + (p(x)). Therefore, p(ex)
= (ao + a ,x + · · · + a11~) + (p(x)) = p(x) + (p(x)) = (p(x)) = 0
Section 39
Introduction to Extension Fields
in F[x]/(p(x)). We have found an element a in E thereforef(a) = 0.
313
= F[x] /(p(x)) such thatp(a) = 0, and +
We illustrate the construction involved in the proof of Theorem 39.3 by two examples.
39.4 Example
Let F = R, and letf(x) = x 2 + 1, which is well known to have no zeros in JR. and thus is irreducible over JR. by Theorem 28.11. Then (x2 + 1) is a maximal ideal in JR.[x], so JR.[x]/(x2 + 1) is a field. Identifying r E JR. with r+ (x2 + 1) in IR.[x] /(x2 + 1), we can view JR. as a subfield of E = IR.[x]/(x2 + 1). Let
a = x + (x 2 + l ).
+ 1), we find 2 2 2 a + l = (x + (x2 + 1)) + (l + (x + 1)) = (x2 + 1) + (x2 + 1) = 0. is a zero of x2 + 1. We shall identify IR. [x]/ (x2 + l ) with C
Computing in JR.[x]/(x2
Thus a of this section.
39.5 Example
near the close .&
and consider f(x) = x 4  5x2 + 6. This time f(x) factors in · · ·
, ya(11)) g(ya(t), · · · , Ya(11))
for f(y 1, · · · ,y11 ), g(y 1, · · · ,y11 ) E F[y 1, · · · ,y11 ], with g(y 1, · · · ,y11 ) ~ 0. It is immediate that a is an automorphism of F(y 1, · · · , y11 ) leaving F fixed. The elements of F(y 1, · · · , y11 ) left fixed by all a, for all a E S,,, are those rational functions that are symmetric in the indeterminates y 1, · · · , Y11 ·
47.1 Definition
An element of the field F(y 1, · · · , y 11 ) is a symmetric function in y 1, · · · , y 11 over F, if • it is fixed by all permutations of y 1, • • • , y11 , in the sense just explained. Let S11 be the group of all the automorphisms a for a E S11 • Observe that S11 is naturally isomorphic to S11 • Let K be the subfield of F(y 1, · · · ,y11 ), which is the fixed field of S11 • Consider the polynomial
n II
f(x) =
(x  y;);
i= I
this polynomial/(x) E (F(y 1, · · · ,y11 ))[x] is a general polynomial of degree n . Let ax be the polynomial extension of a, as defined in Definition 44.4, to (F(y 1, · · · ,y11 )) [x], where ax(x) = x. Now f(x) is fixed by each map ax for a E S11 ; that is,
n(x = n(xII
II
y;)
i= I
Ya(iJ)·
i= I
Thus the coefficients off(x) are in K; they are, except for sign, the elementary symmetric functions in y 1, · · · , y 11 . As illustration, note that the constant term off(x) is ( l)
11
Y 1Y2 · · ·
y,,,
the coefficient of .x' is (y 1 + y 2 + · · · + y 11 ), and so on. These are symmetric functions in y1, · · · ,Y11· The first elementary symmetric function in y 1 , • • • , Y11 is 1 
1
= Y t + Y2 + ... + Y11> the second is s2 = Y1Y2 + Y tY3 + · · · + Y111 Y11, and so on, and the nth is Sn = Y1Y2 · · · Y11 · St
Consider the field E = F(s 1, • • • , s 11 ). Of course, E _::: K , where K is the field of all symmetric function s in y 1, • • • , y11 over F. Since the characteristic of Eis zero, the extension Kover Eis a separable extension. Thus F(y 1 , • • • , y 11 ) is a finite normal extension of E, namely, the splitting field of f(x)
=
ncx
y;)
i= I
over E. Since the degree ofj(x) is n, we have at once that [F(y, , · · · ,y11 ): E] _::: n!
Section 47
Illustrations of Galois Theory
373
(see Exercise 19, Section 44). However, since K is the fixed field of Sn and
ISnl = ISnl = n!, we have also n!
= [F(y1, · · · ,yn) : K ].
Therefore, n! = [F(y,,· ·· , yn): K] .'.S [F(y ,,·· · ,yn): E] .'.Sn!, so
K=E.
The full Galois group of F(y 1, • • • , yn) over Eis therefore Sn. The fact that K = E shows that every symmetric function can be expressed as a rational function of the elementary symmetric functions s 1, • • • ,s,, . We summarize these results in a theorem.
47.2 Theorem
Let F be a field with characteristic zero. Let s 1, • • • , Sn be the elementary symmetric functions in the indeterminates y 1, • • • , Yn· Then every symmetric function of y 1, • • • , y,, over Fis a rational function of the elementary symmetric functions. Also, F(y 1, • • • , y,,) is a finite normal extension of degree n! of F(s 1 , • • • , sn), and the Galois group of this extension is naturally isomorphic to Sn. In view of Cayley's Theorem 8.11, it can be deduced from Theorem 47.2 that any finite group can occur as a Galois group (up to isomorphism). (See Exercise 11.) The proof of Theorem 47 .2 only uses the fact that the characteristic of Fis zero to conclude that the extension F(y 1, y2 , .• • , y,,) over E is a separable extension. With a bit more work, the proof can be modified to allow F to be an arbitrary field .
Examples Let us give our promised example of a finite normal extension having a Galois group whose subgroup diagram does not look like its own inversion.
47.3 Example
K = 1. [H int: odd, what is the order of  t?l
If ~
is a primitive nth root of unity for n
12. Let n, m E z+ be relatively prime. Show that the splitting field in IC of x 11111 splitting field in IC of (x"  l )(x111  l) over Q.

I over Q is the same as the
13. Let n, m E z+ be relatively prime. Show that the group of (x' 1111  1) E Q [x] over Q is isomorphic to the direct product of the groups of (x''  1) E Q [x] and of (x'"  1) E Q [x] over Q . [Hint: Using Galois theory, show that the groups of x'"  l and x''  l can both be regarded as subgroups of the group of xn111  l. Then use Exercises 50 and 5 1 of Section 9.]
384
Part IX
SECTION
49
Galois Theory
INSOLVABILITY OF THE QUINTIC
The Problem We are familiar with the fact that a quadratic polynomial f(x) = ax2 +bx+ c, a op 0, with real coefficients has (b ± .Jb2  4ac)/ 2a as zeros in C. Actually, this is true for f(x) E F[x], where F is any field of characteristic op 2 and the zeros are in F. Exercise 4 asks us to show this. Thus, for example, (x2 + 2x + 3) E