Advanced Algebra I
Homework 11 due on Dec. 22, 2006
(1) Complete the uncompleted proof in the lecture. Especially, the properties of Lagrange resovents.
(2) Let f (x) ∈ Q[x] be a polynomial of degree p, where p is an odd prime. Suppose furthermore that f (x) has only 2 complex roots. Then Galois group of f (x) is Sp.
(3) (*) Construct an example f (x) ∈ Q[x] such that Galois group is A5.
(4) Let p be prime. We consider fa,b: Zp → Zp, given by fa,b(x) = ax + b with a 6= 0. Let
GA(p) := {fa,b|a, b ∈ Zp, a 6= 0}.
(a) GA(p) can be viewed as a subgroup of Sp. It is solvable and transitive of order |GA(p)| = p(p − 1).
(b) (*) A subgroup H < Sp which is solvable and transitive is conjugate to a subgroup of GA(p).
(5) Consider f (x) = xp − x − a ∈ Fp[x]. Is it irreducible for all a ∈ F∗p?
(6) For any give cyclic group Zn, there exists a finite Galois exten- sion F/Q whose Galois group is Zn.
(*) For For any give abelain group G, there exists a finite Galois extension F/Q whose Galois group is G.
(7) Give an example of F/Q such that F is contained in a radical extension, but F/Q is not radical.
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