HW due Fri Apr 27

1. Let F > K be an algebraic field extension, f in F, p(x),q(x) two nonzero polynomials in K[x] of which f is a root. Assume that p(x) has lowest degree. Show that p is irreducible and p|q.


2. Define the derivative q' of a polynomial q without using limits, so that you can apply it to polynomials with arbitrary coefficients.

a. If p,q are two polynomials such that p² | q, show that p divides gcd(q,q').

b. If q is an irreducible polynomial in K[x] where K is a field of characteristic zero, and F is a field extension of K, show that q cannot have a repeated factor when factored in F[x].


3. Let F = F_p(a), the field of rational functions. Consider the map F[x] -> F_p(b) taking x |-> b, a |-> b^p.

a. Prove that x^p-a is irreducible in F[x].
Hint. For r(b) in F_p[b], show that r(b^p) = r(b)^p, a nice extension of the Freshman's Dream. Then reconsider the map above.

b. Let F_p(b) be the extension suggested above, where b^p = a. Show that b is the only root of the irreducible polynomial x^p-a.


4. Let p(x) = the polynomial x^pn - x in F_p[x].

a. Show that p(x) has no repeated roots.

b. Show that p(x) is the product of all monic irreducible polynomials in F_p[x] of degree at most dividing n.

c. Write out this factorization for p=2, n=4.