Compute the following:
a).
b).
To compute these fractions, we ___ ___ _________ ________ _____ ___ ________ ______ _______ ____.
______ _____ ______ _____ _______ ________ __________ ________.
_________ _____ ________.
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Find the elementary divisors and invariant factors of .
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 F is a finite field show that there is always an irreducible polynomial of the form where
.(Hint: Show that
is not a surjective map.)
If , show that a polynomial
is irreducible iff
where
is the group of squares in
.
Describe the set of primes p for which splits into linear factors over
Suppose that is
matrix where A,B,C and D are
matrices. Show that M is symplectic if and only if the following conditions are satisfied:
(Hint: Use block matrix multiplication.) Also, check that the matrix , where A is a
orthogonal matrix, is a symplectic matrix.
Let . Show that G is the cyclic group of order six.