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
.
If , show that a polynomial ax2 + bx + c is irreducible iff
where
is the group of squares in
Find where
.
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 finite extension F of odd degree (greater than 1). Show that
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 .
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.
See Answer →Check that is in the stabiliser of
. Also, show that there are infinitely many choices of
for which
is invertible.
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.)
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 ?
Consider the natural action of on
, the set of
real matrices, by left multiplication.
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 ,
.
Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
See Answer →Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . Sketch the strip and its image.
Prove that the linear fractional transformation maps the circle
into itself. Also prove
that f(z) is conformal in .
Find the zeros and singularities of the function in
. Also find the residue at the poles
Expand in a Laurent series valid for
i) 0 < |z - 1| < 2 and ii) 0 < |z - 3| < 2.