Abstract alegbra Dummit and Foote solution manual PDF

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It is a solution manual for the questions from the book Abstract algebra by Dummit and foote. It contains solutions of some problems from the book....

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Figure 0.1

ABSTRACT ALGEBRA DAVID S. DUMMIT AND RICHARD M. FOOTE

Solutions provided by Scott Larson. Contents 0. Preliminaries 0.1. Basics 0.2. Properties of the Integers 0.3. Z/nZ: The Integers Modulo n Part I – Group Theory 1. Introduction to Groups 1.1. Basic Axioms and Examples 1.2. Dihedral Groups 1.3. Symmetric Groups 1.4. Matrix Groups 1.5. Quaternion Groups 1.6. Homomorphisms and Isomorphisms 1.7. Group Actions 2. Subgroups 2.1. Definition and Examples 2.2. Centralizers and Normalizers, Stabilizers and Kernels 2.3. Cyclic Groups and Cyclic Subgroups 2.4. Subgroups Generated by Subsets of a Group 2.5. Definitions and Examples 2.6. The Lattice of Subgroups of a Group 3. Quotient Groups and Homomorphisms 3.1. Definitions and Examples 3.2. More on Cosets and Lagrange’s Theorem 3.3. The Isomorphism Theorems 3.4. Composition Series and the Holder Program 3.5. Transpositions and the Alternating Group 4. Group Actions 4.1. Group Actions and Permutation Representations 4.2. Groups Acting on Themselves by Left Multiplication – Cayley’s Theorem 4.3. Groups Acting on Themselves by Conjugation – The Class Equation 1

3 3 4 6 8 8 8 10 11 11 13 14 14 15 15 15 16 17 19 20 20 20 21 23 24 24 25 25 25 25

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DAVID S. DUMMIT AND RICHARD M. FOOTE

4.4. Automorphisms 4.5. The Sylow Theorems 4.6. The Simplicity of An 5. Direct and Semidirect Products and Abelian Groups 5.1. Direct Products 5.2. The Fundamental Theorem of Finitely Generated Abelian Groups 5.3. Table of Groups of Small Order 5.4. Recognizing Direct Products 5.5. Semidirect Products 6. Further Topics in Group Theory 6.1. p-groups, Nilpotent Groups, and Solvable Groups Part II – Ring Theory 7. Introduction to Rings 7.1. Basic Definitions and Examples 7.2. Examples: Polynomials Rings, Matrix Rings, and Group Rings 7.3. Ring Homomorphisms and Quotient Rings 7.4. Properties of Ideals 7.5. Rings of Fractions 8. Euclidean, Principal Ideal, and Unique Factorization Domains 8.1. Euclidean Domains 9. Polynomial Rings 9.1. Definitions and Basic Properties 9.2. Polynomial Rings Over Fields I 9.3. Polynomial Rings That are Unique Factorization Domains 9.4. Irreducibility Criteria 9.5. Polynomial Rings Over Fields II 9.6. Polynomials in Several Variables Over a Field and Gr¨obner Bases Part III – Modules and Vector Spaces 10. Introduction to Module Theory 10.1. Basic Definitions and Examples 13. Field Theory 13.1. Basic Theory of Field Extensions 13.2. Basic Theory of Field Extensions 13.3. Classical Straightedge and Compass Constructions 13.4. Splitting Fields and Algebraic Closures 13.5. Separable and Inseparable Extensions 13.6. Cyclotomic Polynomials and Extensions 14. Galois Theory 14.1. Basic Definitions 14.2. The Fundamental Theorem of Galois Theory 15. Commutative Rings and Algebraic Geometry 15.1. Noetherian Rings and Affine Algebraic Sets 15.2. Radicals and Affine Varieties 15.3. Integral Extensions and Hilbert’s Nullstellensatz 15.4. Localization

26 26 27 28 28 29 30 30 30 31 31 33 33 33 34 36 39 42 45 45 46 46 47 48 48 50 50 53 53 53 54 54 54 56 56 56 57 57 57 57 58 58 58 58 58

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0. Preliminaries 0.1. Basics. Proposition 0.1. Let f : A → B . (1) The map f is injective if and only if f has a left inverse. (2) The map f is surjective if and only if f has a right inverse. (3) The map f is a bijection if and only if there exists g : B → A such that f ◦ g is the identity map on B and g ◦ f is the identity map on A. (4) If A and B are finite sets with the same number of elements (|A| = |B|), then f : A → B is bijective if and only if f is injective if and only if f is surjective. Proof. (a) Suppose f is injective so that f −1 (b) contains a single element for b ∈ f (A). Thus g(b) = f −1 (b) is well-defined for b ∈ f (A). Hence g(f (a)) = a for all a ∈ A. Now suppose that f has a left inverse g so that g (f (a)) = a for all a ∈ A. If f (a1 ) = f (a2 ) = b, then g(b) = a1 and g (b) = a2 . But g is a well-defined map so a1 = a2 and thus f is injective. (b) Suppose f is surjective. Take some b ∈ B so we can choose a ∈ A such that f (a) = b. Thus we can define g(b) = a such that f (a) = b. Hence f (g(b)) = f (a) = b. Now suppose that f has a right inverse g so that f (g(b)) = b for all b ∈ B. So f (g(B)) = B and thus f is surjective. (c) Suppose that f is bijective so there is a left inverse, gl , and a right inverse, gr , of f . Let f (a) = b so that a = gl (f (a)) = gl (b) and b = f (gr (b)). Since f is injective, gr (b) = a and thus gl = gr . Thus the right and left inverses of f are the same map. Now suppose that there exists g : B → A such that f ◦ g is the identity map on B and g ◦ f is the identity map on A. Then f is surjective because f (g(B)) = B. Now let f (a1 ) = f (a2 ) so that a1 = g(f (a1 )) = g(f (a2 )) = a2 . Thus f is injective. (d) Suppose that f is injective. Then f takes elements of A to unique elements of B. Since |A| = |B |, f is surjective. Now suppose that f is surjective. Then since |A| = |B|, f takes elements of A to unique elements of B so f is injective. Therefore f is either bijective or neither injective or surjective.  Proposition 0.2. Let A be a nonepty set. (1) If ∼ defines an equivalence relation on A then the set of equivalence classes of ∼ form a partition of A. (2) If {Ai | i ∈ I} is a partition of A then there is an equivalence relation on A whose equivalence classes are preciselly the sets Ai , i ∈ I . In Exercises 1 to 4 let A be the set of 2 × 2 matrices with real number entries. Recall that matrix multiplication is defined by      p q ap + br aq + bs a b = c d r s cp + dr cq + ds

Let

M= and let

  1 1 0 1

B = {X ∈ A | M X = XM } . 0.1.1. The following elements of A lie in B :       1 1 0 0 1 0 , , . 0 1 0 0 0 1 0.1.2. Prove that if P, Q ∈ B, then P + Q ∈ B (where + denotes the usual sum of two matrices). Proof. (P + Q)M = P M + QM = M P + M Q = M (P + Q).  0.1.3. Prove that if P, Q ∈ B, then P · Q ∈ B (where · denotes the usual product of two matrices). Proof. (P · Q)M = P · M · Q = M · P · Q = M (P · Q).



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DAVID S. DUMMIT AND RICHARD M. FOOTE

0.1.4. Find conditions on p, q, r, s which determine precisely when r = 0, p = s.

  p q ∈ B. r s

0.1.5. Determine whether the following functions f are well defined: (1) f : Q → Z defined by f (a/b) = a. (2) f : Q → Q defined by f (a/b) = a2 /b2 . Proof. (a) f is not a well defined map because 1/2 = 2/4 and 1 6= 2. (b) f is a well defined map because if a1 /b1 = a2 /b2 , then a12/b12 = a22 /b22 .

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0.1.6. Determine whether the function f : R+ → Z defined by mapping a real number r to the first digit to the right of the decimal point in a decimal expansion of r is well defined. f is well defined because the decimal expansion of a real number is unique. Thus there is one and only one possible first number to the right of the decimal for each real number. 0.1.7. Let f : A → B be a surjective map of sets. Prove that the relation a ∼ b if and only if f (a) = f (b) is an equivalence relation whose equivalence classes are the fibers of f . Proof. f (a) = f (a) so ∼ is reflexive. If f (a) = f (b), then f (b) = f (a) so ∼ is symmetric. If f (a) = f (b) and f (b) = f (c), then f (a) = f (c) so ∼ is transitive. If a1 , a2 ∈ f −1 (b), then f (a1 ) = f (a2 ) so a1 ∼ a2 . If a1 ∼ a2 then f (a1 ) = f (a2 ) so a1 and a2 are in the same fiber of f .  0.2. Properties of the Integers. (1) (Well Ordering of Z) If A is any nonempty subset of Z+ , there is some element m ∈ A such that m ≤ a, for all a ∈ A. (2) If a, b ∈ Z with a 6= 0, we say a divides b if there is an element c ∈ Z such that b = ac. In this case we write a | b; if a does not divide b we write a ∤ b. (3) If a, b ∈ Z\ {0}, there is a unique positive integer d, called the greatest common divisor of a and b, satisfying: (a) d | a and d | b, (b) if e | a and e | b, then e | d . (4) If a, b ∈ Z\ {0}, there is a unique positive integer l, called the least common multiple of a and B, satisfying: (a) a | l and b | l, (b) if a | m and b | m, then l | m. The connection between the greatest common divisor d and the least common multiple l of the two integers a and b is given by dl = ab. (5) The Division Algorithm: if a, b ∈ Z and b 6= 0, then there exist unique q, r ∈ Z such taht and 0 ≤ r < |b| ,

a = qb + r

where q is the quotient and r the remainder. (6) The Euclidean Algorithm is an important procedure which produces a greatest common divisor of two integers a and b by iterating the Division Algorithm: if a, b ∈ Z\ {0}, then we obtain a sequence of quotients and remainders a = q0 b + r0 b = q1 r0 + r1 r0 = q2 r1 + r2 r1 = q3 r2 + r3 .. . rn−2 = qn rn−1 + rn rn−1 = qn+1 rn where rn is the last nonzero remainder. Such an rn exists since |b| > |r0 | > |r1 | > · · · > |rn | is a decreasing sequence of strictly positive integers if the remainders are nonzero and such a sequence cannot continue indefinitely. Then rn = (a, b).

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(7) One consequence of the Euclidean Algorithm which we chall use regularly is the following: if a, b ∈ Z\ {0}, then there exist x, y ∈ Z such that (a, b) = ax + by that is, the gcd of a and b is a Z-linear combination of a and b. This follows by recursively writing the element rn in the Euclidean Algorithm in terms of the previous remainders. (8) An element p of Z+ is calle a prime if p > 1 and the only posotive divisors of p are 1 and p. An integer n > 1 which is not prime is called composite. An important property of primes is if p is a prime and p | ab, for some a, b ∈ Z then either p | a or p | b. (9) The fundamental theorem of arithmetic says: if n ∈ Z, n > 1, then n can be factored uniquely into the product of primes, i.e., there are distinct primes p1 , p2 , . . . , ps and positive integers α1 , α2 , . . . , αs such that n = pα1 1 p2α2 · · · pαss . This factorization is unique in the sense that if q1 , q2 , . . . , qt are any distinct primes and β1 , β2 , . . . , βt , are positive integers such that n = q1β1 q2β2 · · · q βt t ,

then s = t and if we arrange the two sets of primes in increasing order, then qi = pi and αi = βi , 1 ≤ i ≤ s. Suppose the positive integers a and b are expressed as products of prime powers: a = p1α1 pα2 2 · · · pαs s ,

b = pβ11 pβ2 2 · · · pβss

where p1 , p2 , . . . , ps are distinct and the exponents are ≥ 0. Then the gcd of a and b is min(α1 ,β 1 ) min(α2 ,β 2 ) p2

(a, b) = p1

s ,β s ) · · · pmin(α s

and the lcm is obtained by taking the maximum of the αi and βi instead of the minimum. (10) The Euler ϕ-function is defined as follows: for n ∈ Z+ let ϕ(n) be the number of positive integers a ≤ n with a relatively prime to n, i.e., (a, n) = 1. For primes p, ϕ(p) = p − 1, and, more generally, for all a ≥ 1 we have the formula ϕ(pa ) = pa − pa−1 = pa−1 (p − 1). The function ϕ is multiplicative in the sense that

ϕ(ab) = ϕ(a)ϕ(b) if (a, b) = 1. 1 α2 αs Together with the formula above this gives a general formula for the values of ϕ: if n = pα 1 p 2 · · · p s , then

ϕ(n) = ϕ(p1α1 )ϕ(pα22 ) · · · ϕ(pαs s )

= pα1 1 −1 (p1 − 1)pα22 −1 (p2 − 1) · · · pαs s −1 (ps − 1).

0.2.1. For each of the following paris of integers a and b, determine their gcd, their lcm , and write their gcd in the form ax + by for some integers x and y. (a) (b) (c) (d) (e) (f)

a = 20, b = 13. a = 69, b = 372. a = 792, b = 275. a = 11391, b = 5673. a = 1761, b = 1567. a = 507885, b = 60808.

Solution.

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0.2.2. Prove that if the integer k divides the integers a and b then k divides as + bt for every pair of integers s and t. Proof.

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0.2.3. Prove that if n is composite then there are integers a and b such that n divides ab but n does not divide either a or b. Proof.

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DAVID S. DUMMIT AND RICHARD M. FOOTE

0.2.4. Let a, b and N be fixed integers with a and b nonzero and let d = (a, b) be the gcd of a and b. Suppose x0 and y0 are particular solutions to ax+ by = N . Prove for any integer t that the integers b a x = x0 + t and y = y0 − t d d are also solutions to ax + by = N (this is in fact the general solution). Proof.

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0.2.5. Determine the value ϕ(n) for each integer n ≤ 30 where ϕ denotes the Euler ϕ-function. Solution.

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0.2.6. Prove the Well Ordering Property of Z by induction and prove the minimal element is unique. Proof.

 2

2

0.2.7. If p is a prime prove that there do not exist nonzero integers a and b such that a = pb . Proof.

 +

0.2.8. Let p be a prime, n ∈ Z . Find a formula for the largest power of p which divides n! = n(n−1)(n−2) · · · 2· 1. Solution.

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0.2.9. Write a computer program to determine the greatest common divisor (a, b) of two integers a and b and to express (a, b) in the form ax + by for some integers x and y. 0.2.10. Prove for any given positive integer N there exist only finitely many integers n with ϕ(n) = N where ϕ denotes Euler’s ϕ-function. Conclude in particular that ϕ(n) tends to infinity as n tends to infinity. Proof.

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0.2.11. Prove that if d divides n then ϕ(d) divides ϕ(n) where ϕ denotes Euler’s ϕ-function. Proof.

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0.3. Z/nZ: The Integers Modulo n. Theorem 0.3. The operations of addition and multiplication on Z/nZ are well defined, that is, they do not depend on the choices of representatives for the classes involved. More precisely, if a1 , a2 ∈ Z and b1 , b2 ∈ Z with a1 = b1 and a2 = b2 , then a1 + a2 = b1 + b2 and a1 a2 = b1 b2 , i.e., if a1 ≡ b1

then ×

mod n

a1 + a2 ≡ b1 + b2

and

mod n

a2 ≡ b2

and

Proposition 0.4. Z/nZ) = {a ∈ Z/nZ | (a, n) = 1}.

mod n

a1 a2 ≡ b1 b2

mod n.

0.3.1. Write down explicitly all the elements in the residue classes of Z/18Z. Solution.

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0.3.2. Prove that the distinct equivalence classes in Z/nZ are precisely 0, 1, 2, . . . , n − 1. Proof. 0.3.3. Prove that if a = an 10n + mod 9.

 an−1 n−1

+ · · · + a1 10 + a0 is any positive integer then a ≡ an + an−1 + · · · + a1 + a0

Proof.

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0.3.4. Compute the remainder when 37

100

is divided by 29.

Solution. 0.3.5. Compute the last two digits of 9

 1500

.

Solution. 0.3.6. Prove that the squares of the elements in Z/4Z are just 0 and 1.



7

Proof.

 2

2

0.3.7. Prove for any integers a and b that a + b never leaves a remainder of 3 when divided by 4. Proof.

 2

2

2

0.3.8. Prove that for any integers a and b that a + b = 3c has no solutions in nonzero integers a, b, and c. Proof.

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0.3.9. Prove that the square of any odd integer always leaves a remainder of 1 when divided by 8. Proof.

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0.3.10. Prove that the number of elements of (Z/nZ)× is ϕ(n) where ϕ denotes the Euler ϕ-function. Proof.

 ×

×

0.3.11. Prove that if a, b ∈ (Z/nZ) , then a · b ∈ (Z/nZ) . Proof.

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0.3.12. Let n ∈ Z, n > 1, and let z ∈ Z with 1 ≤ a ≤ n. Prove if a and n are not relatively prime then there exists an itneger b with 1 ≤ b < n such that ab ≡ 0 mod n and deduce that there cannot be an itneger c such that ac ≡ 1 mod n. Proof.

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0.3.13. Let n ∈ Z, n > 1, and let a ∈ Z with 1 ≤ a ≤ n. Prove that if a and n are relatively prime then there is an integer c such that ac ≡ 1 mod n. Proof.

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0.3.14. Conclude from the previous two exercises that (Z/nZ)× is the set of elements a of Z/nZ with (a, n) = 1 and hence prove proposition 0.4. Verify this directly in the case n = 12. Solution.

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0.3.15. For each of the following pairs of integers a and n, show that a is relatively prime to n and determine the multiplicative inverse of a in Z/nZ. (a) a = 13, n = 20. (b) a = 69, n = 89. (c) a = 1891, n = 3797. (d) a = 6003722857, n = 77695236973. Proof.

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0.3.16. Write a computer program to add and multiply mod n, for any n given as input. The output of these operations should be the least residues of the usms and products of two integers. Also include the feature that if (a, n) = 1, and integer c between 1 and n − 1 such taht a · c = 1 may be printed on request.

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DAVID S. DUMMIT AND RICHARD M. FOOTE

Part I – Group Theory 1. Introduction to Groups 1.1. Basic Axioms and Examples. Proposition 1.1. If G is a group under the operation ⋆, then (1) the identity of G is unique (2) for each a ∈ G, a−1 is uniquely determined (3) (a−1 )−1 = a for all a ∈ G (4) (a ⋆ b)−1 = (b−1 ) ⋆ (a−1 ) (5) for any a1 , a2 , . . . , an ∈ G the value of a1 ⋆ a2 ⋆ · · · ⋆ an is independent of how the expression is bracketed. Proposition 1.2. Let G be a group and let a, b ∈ G. The equations ax = b and ya = b have unique solutions for x, y ∈ G. In particular, the left and right cancellation laws hold in G, i.e., (1) if au = av, then u = v, (2) if ub = vb, then u = v. Let G be a group. 1.1.1. Determine which of the following binary operations are associative: (a) the operation ⋆ on Z defined by a ⋆ b = a − b (b) the operation ⋆ on R defined by a ⋆ b = a + b + ab (d) the operation ⋆ on Z × Z defined by (a, b) ⋆ (c, d) = (ad + bc, bd). Note that (1 − 2) − 3 = −4 and 1 − (2 − 3) = 2, so (a) is not associative. We calculate that (b) and (d) are associative below. (a ⋆ b) ⋆ c = (a + b + ab) + c + (a + b + ab)c = a + b + c + ab + ac + bc + abc = a + b + c + bc + a(b + c + bc) = a ⋆ (b ⋆ c)     (a, b) ⋆ (c, d) ⋆ (e, f ) = (ad + bc)f + bde, bdf   = adf + b(cf + de), bdf   = (a, b) ⋆ (c, d) ⋆ (e, f )

1.1.2. Decide which of the following binary operations are commutative: (a) the operation ⋆ on Z defined by a ⋆ b = a − b (b) the operation ⋆ on R defined by a ⋆ b = a + b + ab (d) the operation ⋆ on Z × Z defined by (a, b) ⋆ (c, d) = (ad + bc, bd). Note that 1 − 2 = −1 and 2 − 1 = 1 so (a) is not commutative. We calculate that (b) and (d) are commutative below. a ⋆ b = a + b + ab = b + a + ba =b⋆a (a, b) ⋆ (c, d) = (ad + bc, bd ) = (cb + da, db) = (c, d) ⋆ (a, b) 1.1.3. Prove that addition of residue classes in Z/nZ is associative. Proof.

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1.1.4. Prove that multiplication of residue classes in Z/nZ is associative. Proof.

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1.1.5. Prove for all n > 1 that Z/nZ is not a group under multiplication of residue classes. Proof.

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1.1.6. Determine which of the following sets are groups under addition: (b) the set of rational numbers in lowest terms whose denominators are even together with 0 (d) the set of rational numbers of absolute value ≥ 1 together with 0 (e) the set of rational numbers with denominators equal to 1 or 2. Note that 1/6 + 1/6 = 1/3 so addition is not a binary relation on (b) and thus (b) is not a group. Also note that −3/2 + 1 = −1/2 so addition is not a binary relation on (d) and thus (d) is note a group. We notice that addition is a well defined binary relation on (e), addition is associative on the rational numbers so this subset must also have the associative property, the identity is 0, and the inverse of a is −a. Therefore (e) is a group. 1.1.7. Let G = {x ∈ R | 0 ≤ x < 1} and for x, y ∈ G let x⋆ y be the fractional part of x+y (i.e., x ⋆y = x+y −[x+y] where [a] is the greatest integer less than or equal to a). Prove that ⋆ is a well defined binary operation on G and that G is an abelian group under ⋆ (called the real numbers mod 1). Proof. Note that the sum of two real numbers has a unique fractional part, x, such that 0 ≤ x < 1. Thus ⋆ is a well defined operation that takes ...