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Final Examination MATH 546/701I Section 001 15 December 2001
Problem 0.
Let m, n, and a be positive integers. Suppose that (m, n) = 1 and m | a and n | a.
Prove that mn | a.
Problem 1: Core Let Θ be the set of ordered pairs of integers described by:
〈a, b〉 ∈ Θ if and only if there is some integer c so that 2 | (a − c) and 3 | (b − c).
Prove that Θ is a symmetric relation of the set of all integers.
Problem 2: Core Let A, B, and C be sets. Let h be a function from A onto B and let g be a function from A onto C. Let
f = {〈h(a), g(a)〉 | a ∈ A}.
Suppose that f is a function from B into C. Prove KER h ⊆ KER g.
Problem 3.
Let k be an integer with k > 1 and let h be a homomorphism from 〈Z, +, ·, −, 0 , 1 〉 onto 〈Zk, +k, ·k, −k, 0 , 1 〉. Prove that for any integers m and n, if h(m) = h(n) = 0, then (m, n) 6 = 1.
Problem 4: Core Let G be a group, let H be a subgroup of G, and let g ∈ G. Define K = {g−^1 hg | h ∈ H}. Prove that K is also a subgroup of G.
Problem 5. Let R be the set of all real numbers. Let F be the set of all functions from R into R. Define addition, negation, and the zero function 0 as follows:
(f + g)(r) = f (r) + g(r) (−f )(r) = −f (r) 0 (r) = 0
for all f, g ∈ F and all real numbers r. With these operations, all familiar from calculus classes, 〈F, +, −, 0 〉 becomes a group. (You are not asked to prove that here.) Define Φ : F → R via Φ(f ) = f (π)
for all f ∈ F. Prove that Φ is a homomorphism from 〈F, +, −, 0 , 〉 into 〈R, +, −, 0 〉.
Problem 6. Determine in each part below whether the given permutation is even or odd. Please explain your reasoning. a. (1, 2)(3, 2 , 1) b. (1, 2 , 3)(1, 2 , 3 , 4 , 5)(4, 5) c. (2, 5)(2, 3)(2, 4)(2, 1)(2, 5)
Problem 7: Core Let A, B, and C be groups. Let h be a homomorphism from B onto A and let g be a homomorphism from C onto A. 1
2
Prove that there is an isomorphism α from B/ KER h onto C/ KER g. [Warning! Check the directions of the maps g and h.]
Problem 8. Let F be a field and let f (x), g(x), and h(x) be polynomials with coefficients from F. Prove that if (f (x), g(x)) = 1 and f (x) | h(x) and g(x) | h(x), then f (x)g(x) | h(x).
Problem 9. Let F be a field and let a(x), b(x), c(x) ∈ F[x]. Suppose that ϕ and ψ are homomorphisms from F[x] into F such that
ϕ(a(x)) = ϕ(c(x)) and ψ(b(x)) = ψ(c(x))
Find a polynomial d(x) ∈ F[x] so that
ψ(a(x)) = ψ(d(x)) and ϕ(b(x)) = ϕ(d(x))
Problem 10. Let F be a field. Let F denote the set of all functions from F into F. Define Addition, multiplication, negation, and the constant functions as follows:
(f + g)(r) = f (r) + g(r) (f • g)(r) = f (r) · g(r) (−f )(r) = −f (r) 0 (r) = 0 1 (r) = 1
for all f, g ∈ F and all r ∈ F. With these operations 〈F, +, −, • , 0 , 1 〉 becomes a ring. (You are not asked to prove that here.) Define Φ : F[x] → F by letting Φ(p(x)) be the function denoted by the polynomial p(x) ∈ F[x]. So if Φ(p(x)) = p, then p(r) is the result of evaluating p(x) at r, for every r ∈ F and every p(x) ∈ F[x]. Prove that Φ is a homomorphism. Prove this homomorphism is one-to-one if F is infinite.
Problem 11. In each part below determine whether the given polynomial is an iireducible member of Q[x].
a. x^4 − 4 x^2 + 2. b. x^4 − 4 x^2 + 4. c. x^4 + 5.
Extra Credit Let G be a group and let H, K, and N be subgroups of G fulfilling all the following conditions:
- H ⊆ K,
- HN = KN ,
- H ∩ N = K ∩ N.
Under these stipulations, prove that H = K.
