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MATH 3613 Homework Set 1 1. Prove that “not-Q ⇒ not-P ” implies “P ⇒ Q” 2. Prove that if m and n are even integers, then n + m is an even integer. 3. Prove that if n is an odd integer, then n2 is an odd integer. 4. Prove that if n is an integer and n2 is odd, then n is odd. 5. Prove, by the contrapositive method, that if c is an odd integer then the equation n2 + n − c = 0 has no integer solution. 6. Prove, by mathematical induction, that if n ≥ 5 then 2n > n2 . 7. Prove by mathematical induction that n X n(n + 1)(2n + 1) i2 = 6 i=1 , ∀ n ∈ Z+ 8. Prove the following identities. (a) B ∩ (C ∪ D) = (B ∩ C) ∪ (B ∩ D) (b) B ∪ (C ∩ D) = (B ∪ C) ∩ (B ∪ D) (c) C = (C − A) ∪ (C ∩ A) 9. Describe each set in set-builder notation: (a) All positive real numbers. (b) All negative irrational numbers. (c) All points in the coordinate plane with rational first coordinate. (d) All negative even integers greater than −50. 10. Which of the following sets are nonempty? (a) r ∈ Q | r2 = 2 (b) r ∈ R | r2 + 5r − 7 = 0 (c) t ∈ Z | 6t2 − t − 1 = 0 1 . 2 11. Is B is a subset of C when (a) B = Z and C = Q? (b) B = all solutions of x2 + 2x − 5 = 0 and C = Z? (c) B = {a, b, 7, 9, 11, −6} and C = Q? 12. In each part find B − C, B ∩ C, and B ∪ C. (a) B = Z and C = Q. (b) B = R and C = Q. (c) B = {a, b, c, 1, 2, 3, 4, 5, 6} and C = {a, c, e, 2, 4, 6, 8}. 13. Let A, B be subsets of U . Prove De Morgan’s laws: (a) U − (A ∩ B) = (U − A) ∪ (U − B) (b) U − (A ∪ B) = (U − A) ∩ (U − B) 14. (a) Give an example of a function that is injective but not surjective. (b) Give and example of a function that is surjective but not injective. 15. Prove that f : R → R : f (x) = 2x + 1 16. Prove that f : R → R : f (x) = −3x + 5 is injective. is surjective. 17. Let B and C be nonempty sets. Prove that the function f :B×C → C ×B given by f (x, y) = (y, x) is a bijection. 3 Homework Set 2 (Homework Problems from Chapter 1) Problems from Section 1.1. 1.1.1. Let n be an integer. Prove that a and c leave the same remainder when divided by n if and only if a − c = nk for some k ∈ Z. 1.1.2, Let a and b be integers with c 6= 0. Then there exist unique integers q and r such that (i) a = cq + r (ii) 0 ≤ r < |c| . 1.1.3. Prove that the square of any integer a is either of the form 3k or of the form 3k + 1 for some integer k. 1.1.4. Prove that the cube of any integer has exactly one of the forms 9k, 9k + 1, or 9k + 8. Problems from Section 1.2 1.2.1. (a) Prove that if a | b and a | c then a | (b + c). (b) Prove that if a | b and a | c, then a | (br + ct) for any r, t ∈ Z. 1.2.2. Prove or disprove that if a | (b + c), then a | b or a | c. 1.2.3. Prove that if r ∈ Z is a non-zero solution of x2 + ax + b = 0 (where a, b ∈ Z), then r | b. 1.2.4. Prove that GCD(a, a + b) = d if and only if GCD(a, b) = d. 1.2.5. Prove that if GCD(a, c) = 1 and GCD(b, c) = 1, then GCD(ab, c) = 1. 1.2.6. (a) Prove that if a, b, u, v ∈ Z are such that au + bv = 1, then GCD(a, b) = 1. (b) Show by example that if au + bv = d > 0, then GCD(a, b) need not be d. Problems from Section 1.3 1.3.1. Let p be an integer other than 0, ±1. Prove that p is prime if and only if for each a ∈ Z, either GCD(a, p) = 1 or p | a. 1.3.2 Let p be an integer other than 0 ± 1 with this property: Whenever b and c are integers such that p | bc, then p | c or p | b. Prove that p is prime. 1.3.3. Prove that if every integer integer n > 1 can be written in one and only one way in the form n = p1 p2 · · · pr where the pi are positive primes such that p1 ≤ p2 ≤ · · · ≤ pr . 1.3.4. Prove that if p is prime and p | an , then pn | an . 1.3.5. (a) Prove that there exist no nonzero integers a, b such that a2 = 2b2 . 4 (b) Prove that √ 2 is irrational. 5 Homework Set 3 (Problems from Chapter 2) Problems from §2.1 2.1.1. Prove that a ≡ b (mod n) if and only if a and b leave the same remainder when divided by n. 2.1.2. If a ∈ Z, prove that a2 is not congruent to 2 modulo 4 or to 3 modulo 4. 2.1.3. If a, b are integers such that a ≡ b (mod p) for every positive prime p, prove that a = b. 2.1.4. Which of the following congruences have solutions: (a) x2 ≡ 1 (mod 3) (b) x2 ≡ 2 (mod 7) (c) x2 ≡ 3 (mod 11) 2.1.5. If [a] = [b] in Zn , prove that GCD(a, n) = GCD(b, n). 2.1.6. If GCD(a, n) = 1, prove that there is an integer b such that ab = 1 (mod n). 2.1.7. Prove that if p ≥ 5 and p is prime, then either [p]6 = [1]6 or [p]6 = [5]6 . Problems from §2.2 2.2.1. Write out the addition and multiplication tables for Z4 . 2.2.2. Prove or disprove: If ab = 0 in Zn , then a = 0 or b = 0. 2.2.3. Prove that if p is prime then the only solutions of x2 + x = 0 in Zp are 0 and p − 1. 2.2.4. Find all a in Z5 for which the equation ax = 1 has a solution. 2.2.5. Prove that there is no ordering ≺ of Zn such that (i) (ii) if a ≺ b, and b ≺ c, then a ≺ c; if a ≺ b, then a + c ≺ b + c for every c ∈ Zn . Problems from §2.3 2.3.1 If n is composite, prove that there exists a, b ∈ Zn such that a 6= [0] and b 6= [0] but ab = [0]. 2.3.2 Let p be prime and assume that a 6= 0 in Zp . Prove that for any b ∈ Zp , the equation ax = b has a solution. 2.3.3. Let a 6= [0] in Zn . Prove that ax = [0] has a nonzero solution in Zn if and only if ax = [1] has no solution. 2.3.4. Solve the following equations. (a) 12x = 2 in Z19 . (b) 7x = 2 in Z24 . (c) 31x = 1 in Z50 . (d) 34x = 1 in Z97 . 6 Homework Set 4 (Problems from Chapter 3) §3.1 3.1.1. The following subsets of Z (with ordinary addition and multiplication) satisfy all but one of the axioms for a ring. In each case, which axiom fails. (a) The set S of odd integers. (b) The set of nonnegative integers. 3.1.2. (a) Show that the set R of all multiples of 3 is a subring of Z. (b) Let k be a fixed integer. Show that the set of all multiples of k is a subring of Z. 3.1.3. Let R = {0, e, b, c} with addition and multiplication defined by the tables below: + 0 0 0 e e b b c c e e 0 c b b b c 0 e · 0 e b c c c b e 0 0 0 0 0 0 e 0 e b c b 0 b e c c 0 c c 0 Assume distributivity and associativity and show that R is a ring with identity. Is R commutative? 3.1.4. Let F = {0, e, a, b} with addition and multiplication defined by the tables below: + 0 e a b 0 0 e a b e e 0 b a · 0 e a b a b a b b a 0 e e 0 0 0 0 0 0 e 0 e a b a b 0 0 a b b e e a Assume distributivity and associativity and show that R is a field. 3.1.5. Which of the following five sets are subrings of M (R). Which ones have an identity? (a) A= (b) B= a 0 0 0 (c) C= (d) D= (e) D= |r∈Q b c a b r 0 | a, b, c ∈ Z a b | a, b ∈ R a a 0 0 a 0 0 a |a∈R |a∈R 3.1.6. Let R and S be rings. Show that the subset R̄ = {(r, 0S ) | r ∈ R} is a subring of R × S. 3.1.7 If R is a ring, show that R∗ = {(r, r) | r ∈ R} is a subring of R × R. 7 3.1.8. Is {1, −1, i, −i} a subring of C? 3.1.9. Let p be a positive prime and let R be the set of all rational numbers that can be written in the form r pi with r, i ∈ Z. Show that R is a subring of Q. 3.1.10. Let T be the ring of continuous functions from R to R and let f, g be given by 0 if x ≤ 2 2−x if x ≤ 2 f (x) = , g(x) = x−2 if 2 < x 0 if 2 < x . Show that f, g ∈ T and that f g = 0T , and therefore that T is not an integral domain. 3.1.11. Let n o √ √ Q( 2) = r + s 2 | r, s ∈ Q . √ Show that Q( 2) is a subfield of R. 3.1.12. Prove Theorem 3.1: If R and S are rings, then the Cartesian product R × S can be given the structure of a ring by setting (r, s) + (r0 , s0 ) = (r + r0 , s + s0 ) (r, s)(r0 , s0 ) = (rr0 , ss0 ) 0R×S = (0R , 0S ) . Also, if R and S are both commutative, then so is R × S; and if R and S each have an identity, then so does R × S. 3.1.13. Prove or disprove: If R and S are integral domains, then R × S is an integral domain. 3.1.14. Prove or disprove: If R and S are fields, then R × S is a field. §3.2 3.2.1. If R is a ring and a, b ∈ R then (a) (a + b)(a − b) =? (b) (a + b)3 =? (c) What are the answers to (a) and (b) if R is commutative? 3.2.2. An element e of a ring R is said to be idempotent if e2 = e. (a) Find four idempotent elements of the ring M2 (R). (b) Find all idempotents in Z12 . 3.2.3. Prove that the only idempotents in an integral domain R are 0R and 1R . 3.2.4. Prove or disprove: The set of units in a ring R with an identity is a subring of R. 3.2.5. (a) If a and b are units in a ring R with identity, prove tha ab is a unit and (ab)−1 = b−1 a−1 . (b)Give an example to show that if a and b are units, then (ab)−1 may not be the same as a−1 b−1 . (Hint: consider the matrices i and k in the quaterion ring H.) 3.2.6. Prove that a unit in a commutative ring cannot be a zero divisor. 3.2.7. (a) If ab is a zero divisor in a commutative ring R, prove that a or b is a zero divisor. (b) If a or b is a zero divisor in a commutative ring R and ab 6= 0R , prove that ab is a zero divisor. 8 3.2.8. Let S be a non-empty subset of a ring R. Prove that S is a subring if and only if for all a, b ∈ S, both a − b and ab are in S. 3.2.9. Let R be a ring with identity. If there is a smallest integer n such that n1R = 0R , then n is said to have characteristic n. If no such n exists, R is said to have characteristic zero. Show that Z has characteristic zero, and that Zn has characteristic n. What is the characteristic of Z4 × Z6 ? §3.3 3.3.1. Let R be a ring and let R∗ be the subring of R × R consisting of all elements of the form (a, a), a ∈ R. Show that the function f : R → R∗ given by f (a) = (a, a) is an isomorphism. 3.3.2. If f : Z → Z is an isomorphism, prove that f is the identity map. 3.3.3. Show that the map f : Z → Zn given by f (a) = [a] is a surjective homomorphism but not an isomorphism. 3.3.4. If R and S are rings and f : R → S is a ring homomorphism, prove that f (R) = {s ∈ S | s = f (a) for somea ∈ R} is a subring of S 3.3.5. (a) If f : R → S and g : S → T are ring homomorphisms, show that g ◦ f : R → T is a ring homomorphism. (b) If f : R → S and g : S → T are ring isomorphisms, show that g ◦ f : R → T is also a ring isomorphism. 3.3.6. If f : R → S is an isomorphism of rings, which of the following properties are preserved by this isomorphism? Why? (a) a ∈ R is a zero divisor. (b) R is an integral domain. (c) R is a subring of Z. (d) a ∈ R is a solution of x2 = x. (e) R is a ring of matrices. 3.3.7. Use the properties that are preserved by ring isomorphism to show that the first ring is not isomorphic to the second. (a) E (the set of even integers) and Z. (b) R × R × R × R and M2 (R). (c) Z4 × Z14 and Z16 . (d) Q and R. (e) Z × Z2 and Z. (f) Z4 × Z4 and Z16 . 9 Homework Set 5 (Problems from Chapter 4) §4.1 4.1.1. Perform the indicated operations in Z6 [X] and simply your answer. (a) (3x4 + 2x3 − 4x2 + x − 4) + (4x3 + x2 + 4x + 3) (b) (x + 1)3 4.1.2. Which of the following subsets of R[x] are subrings of R[x]? Justify your answer. (a) S = {All polynomials with constant term 0R }. (b) S = {Alll polynomials of degree 2 }. (c) S = {All polynomials of degree ≤ k ∈ N, where 0 < k}. (d) S = {All polynomials in which odd powers of x have zero coefficients}. (e) S = {All polynomials in which even powers of x have zero coefficients}. 4.1.3. List all polynomials of degree 3 in Z2 [x]. x3 , x3 + 1, x3 + x, x3 + x + 1, x3 + x2 , x3 + x2 + 1, x3 + x2 + x, x3 + x2 + x + 1 4.1.4. Let F be a field and let f (x) be a non-zero polynomial in F [x]. Show that f (x) is a unit in F [x] if and only if deg f (x) = 0. §4.2 4.2.1. If a, b ∈ F and a 6= b, show that x + a and x + b are relatively prime in F [x]. 4.2.2. Let f (x), g(x) ∈ F [x]. If f (x) | g(x) and g(x) | f (x), show that f (x) = cg(x) for some non-zero c ∈ F . (b) If f (x) and g(x) are monic and f (x) | g(x) and g(x) | f (x), show that f (x) = g(x). 4.2.3. Let f (x) ∈ F [x] and assume f (x) | g(x) for every nonconstant g(x) ∈ F [x]. Show that f (x) is a constant polynomial. 4.2.4. Let f (x), g(x) ∈ F [x], not both zero, and let d(x) = GCD (f (x), g(x)). If h(x) is a common divisor of f (x) and g(x) of highest possible degree, then prove that h(x) = cd(x) for some nonzero c ∈ F . 4.2.5. If f (x) is relatively prime to 0F , what can be said about f (x). 4.2.6. Let f (x), g(x), h(x) ∈ F [x], with f (x) and g(x) relatively prime. If f (x) | h(x) and g(x) | h(x), prove that f (x)g(x) | h(x). 4.2.7. Let f (x), g(x), h(x) ∈ F [x], with f (x) and g(x) relatively prime. If h(x) | f (x), prove that h(x) and g(x) are relatively prime. 4.2.8. Let f (x), g(x), h(x) ∈ F [x], with f (x) and g(x) relatively prime. Prove that the GCD of f (x)h(x) and g(x) is the same as the GCD of h(x) and g(x). 10 §4.3 4.3.1 Prove that f (x) and g(x) are associates in F [x] if and only if f (x) | g(x) and g(x) | f (x). 4.3.2 Prove that f (x) is irreducible in F [x] if and only if its associates are irreducible. 4.3.3. If p(x) and q(x) are nonassociate irreducibles in F [x], prove that p(x) and q(x) are relatively prime. §4.4 4.4.1. Verify that every element of Z3 is a root of f = x3 − x ∈ Z3 . 4.4.2. Use the Factor Theorem to show that f = x7 − x factors in Z7 as f = x (x − [1]7 ) (x − [2]7 ) (x − [3]7 ) (x − [4]7 ) (x − [5]7 ) (x − [6]7 ) . 4.4.3. If a ∈ F is a nonzero root of f = cn xn + . . . + c1 x + c0 ∈ F [x] , show that a−1 is a root of g = c0 xn + c1 xn−1 + · · · + cn . 4.4.4. Prove that x2 + 1 is reducible in Zp [x] if and only if there exists integers a and b such that p = a + b and ab ≡ 1 (mod p). 4.4.5. Find a polynomial of degree 2 in Z6 [x] that has four roots in Z6 . Does this contradict Corollary 4.13?