Let and let
be algebraic over F of degree m and n, respectively. Show that
. What can you say about
if m and n are coprime?
Let (F \subset K) and (\alpha, \beta \in K) be algebraic over (F) of degrees (m) and (n), respectively. That is, the minimal polynomial of (\alpha) over (F) has degree (m) and that of (\beta) over (F) has degree (n). _________ _____ _______ ______ __________ _______ ____ ____ __________ __________ _________ ___.
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Let . Show that G is the cyclic group of order six.
(b) Solve the following set of congruences:
(c) Show that is not a UFD by giving two different factorisations of 20.
Suppose that in the remaining parts of this exercise. We will show that the stabiliser of
is infinite. If
, the stabiliser of
is
. So suppose
. Let us write
. Then,
for non-zero
. Why ?
By looking at the factorisation of guess the number of irreducible polynomials of degree 2 over
. Find all the irreducible polynomials of degree 2 over
.
Consider the natural action of on
, the set of
real matrices, by left multiplication.
Describe the set of primes p for which x2 - 11 splits into linear factors over .
The aim of this exercise is to show that acts transitively on
In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group