Which of the following statements are true and which are false? Give reasons for your answer.
(a) If a finite group G acts on a finite set S, then for all
.
(b) There are exactly 8 elements of order 3 in S4.
(c) If , then
.
(d) .
(e) For any ,
.
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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 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.)
Let p be an odd prime that divides . Suppose
. Show that C is the unique Sylow p-subgroup of Dn.
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
.
Which of the following statements are true and which are false? Give reasons for your answer.
(a) If a finite group G acts on a finite set S, then for all
.
(b) There are exactly 8 elements of order 3 in S4.
(c) If , then
.
(d) .
(e) For any ,
.
) Prove the relation
Further, find all the elements of order 2 in Dn.
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.)