If is a finite field show that there is always an irreducible polynomial of the form x3 - x + a where
. (Hint: Show that
is not a surjective map.)
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) Let be a finite extension F of odd degree (greater than 1). Show that
Let be a vector that is not a scalar multiple of
. Show that there is a matrix
such that
and
. (Hint: Set up two sets of simultaneous equations in two unknowns and argue why they have a solution.)
The aim of this exercise is to show that acts transitively on
Find the elementary divisors and invariant factors of .
Let H be a finite group and, for some prime p, let P be a p-Sylow subgroup of H which is normal in H. Suppose H is normal in K, where K is a finite group. Then, show that P is normal in K.
Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
(b) Is there a finite group with class equation 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2?
(c) Compute the following: a)
b)
Describe the set of primes p for which x2 - 11 splits into linear factors over .