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?
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Describe the set of primes p for which x2 - 11 splits into linear factors over .
Suppose n is even, , where
. Let
and
. Show that HN is a subgroup of Dn. What is its order?
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)
If , show that a polynomial ax2 + bx + c is irreducible iff
where
is the group of squares in
) Prove the relation
Further, find all the elements of order 2 in Dn.
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 ,
.
Consider the natural action of on
, the set of
real matrices, by left multiplication.