Homework for Week 12
Part 1
- (Rotman 3.29 ii-iii) True or false, with explanation.
- If R is a domain then R[x] is a domain.
- Let Q denote the rational numbers. Q[x] is a field.
- (Rotman 3.30) Show that if R is a nonzero commutative ring, then R[x] is never a field.
- (Rotman 3.46) Let R be a commutative ring.
Show that the function f: R[x] → R given by
f(a0 + a1 x + a2x2 + ...)
= a0 is a homomorphism.
Describe ker f in terms of roots of polynomials.
What does the first isomorphism theorem say in this context?
- Suppose that R is a commutative domain and a is an element of R.
Prove that the map f: R[x] → R defined by f(p(x)) = p(a) is a
homomorphism. Describe in words the kernel ker(f).
What does the first isomorphism theorem say here?
- Suppose that f: C[x] → C is
the homomorphism defined by f(p(x)) = p(1). (C is the complex numbers.)
Describe
in words the kernel ker(f).
What does the first isomorphism theorem say here?
Part 2
- (Rotman 3.56i-iv) True or false with explanation.
- If a(x), b(x) are polynomials in (Z/5Z)[x] with b(x) nonzero, then there are polynomials c(x), d(x) in (Z/5Z)[x] with a(x)=b(x)c(x)+d(x), where either d(x)=0 or deg(d(x)) < deg(b(x)).
- If f(x), g(x) are polynomials in Z[x] with g(x) nonzero, then there are polynomials q(x), r(x) in Z[x] with f(x)=g(x)q(x)+r(x), where either r(x)=0 or deg(r(x)) < deg(g(x)).
- The gcd of 2x2+4x+2 and 4x2+12x+8 in Q[x] is 2x+2.
- If R is a domain then every unit in R[x] has degree 0.
- Find the gcd of x2 - x - 2 and x3-7x+6 in Q[x] and express the gcd as a linear combination of the two polynomials.
- (Rotman 3.58) Find the gcd of x2 - x - 2 and x3-7x+6 in (Z/5Z)[x] and express the gcd as a linear combination of the two polynomials.
- (Rotman 3.59) Let k be a field and let f(x) be a nonzero polynomial in k[x]. Suppose that a1, a2, a3, ..., am are distinct roots of f(x) in k[x]. Prove that f(x) = (x-a1)(x-a2)(x-a3)...(x-a1)g(x) for some g(x) in k[x].
- (Rotman 3.61) Let R be an arbitrary commutative ring. Suppose that f(x) is a polynomial in R[x] and a in R is a root of f(x), so f(a)=0. Prove that there is a factorization f(x)=(x-a)g(x) in R[x].
- Part 1: Due Tuesday 4/17
- Part 2: Due Friday 4/20