MA 3419: Galois theory
Homework problems due December 10, 2015
Solutions to this are due by the end of the class on Thursday December 10. Please attach
a cover sheet with a declaration http://tcd-ie.libguides.com/plagiarism/declaration
confirming that you know and understand College rules on plagiarism.
Please put your name and student number on each of the sheets you are handing in. (Or,
ideally, staple them together).
1. (a) Show that if Fpn is isomorphic to an extension of F(p 0 )n 0 (where p and p 0 are primes)
then p = p 0 and n is divisible by n 0 .
(b) Explain why Fpn is a Galois extension of Fp .
(c) Show that the Galois group Gal(Fpn : Fp ) is the cyclic group Z/nZ. (Hint: show that
x 7→ xp is an automorphism, and that it is of order n in the Galois group).
(d) Show that if n is divisible by n 0 then Fpn is isomorphic to a field extension of Fpn 0 .
(e) Show that if n is divisible by n 0 then Fpn is a Galois extension of Fpn 0 , and describe
the Galois group Gal(Fpn : Fpn 0 ).
2. Explain how to compute cos(2π/13) by solving quadratic and cubic equations only.
In the following problem, you may use without proof the following results:
• If n = pa1 1 · · · pak k where pi are distinct primes, we have
∼ (Z/pa1 Z)× × · · · × (Z/pak Z)× .
(Z/nZ)× =
1
k
∼ Z/pk−1 (p − 1)Z.
• If p is an odd prime, we have (Z/pk Z)× =
k
×
k−2
∼
• We have (Z/2 Z) = Z/2 Z × Z/2Z for k > 3.
3. Show that there exist some complex numbers roots of unity ξ1 , . . . , ξs , and
some rational numbers a1 , . . . , as , so that the number α = a1 ξ1 + · · · + as ξs satisfies
∼ Z/720Z.
Gal(Q(α) : Q) =
4. (a) Considering x5 − x − 1 over F2 and over F5 , establish that the Galois group of the
splitting field of x5 − x − 1 over Q is S5 .
(b) Let p be a prime number. Show that for large N the polynomial
xp − N3 p3 x(x − 1) · · · (x − (p − 4)) − p
has p − 2 real roots. Use it to deduce that for such values of N the Galois group of the
splitting field of this polynomial over Q is Sp .