Math 6320, Assignment 1
Due in class: Tuesday, February 2
In the problems below, K denotes a field; F p denotes the field with p elements.
1. Prove that K[x] contains infinitely many monic irreducible polynomials.
2. If [K(a) : K] is an odd integer, prove that K(a2 ) = K(a).
3. Let a ∈ K(x) r K. Prove that K(x) is an algebraic extension of the field K(a).
4. If f (x) ∈ Q[x] is irreducible with roots α1 , . . . , αn ∈ C, prove that αi − α j ∈
/ Q for all i 6= j.
5. Let K ⊂ L be an algebraic extension of fields. Prove that each irreducible polynomial in L[x] has a nonzero
multiple in K[x].
6. Let ϕ(n) be the Euler ϕ-function. Let n be a positive integer. Show that n = ∑d|n ϕ(d), where the sum is taken
over all positive integers d that divide n.
7. Let G be a finite group with identity e, with the property that for each integer d > 1, the equation xd = e has at
most d solutions in G. Prove that G is cyclic.
Conclude that the multiplicative group of a finite field is cyclic.
8. Let K have characteristic p > 0. If x p − x − a is reducible in K[x], prove it splits in K[x].
9. For each of the polynomials below, determine the extension degree of its splitting field:
(a) x5 − 7 ∈ Q[x]
(b) x8 − 2 ∈ Q[x]
(c) x p − x − 1 ∈ F p [x]
10. Let F be a finite field. Prove that any element of F can be written as a sum of two squares.