Show that is not a UFD by giving two different factorisations of 20
To show that the ring of integers of \( \mathbb{Q}(\sqrt{-19}) \) is not a Unique Factorization Domain (UFD), we need to find two different factorizations of an element in the ring. Let's consider the element \( 20 \).
We know that the ring of integers of \( \mathbb{Q}(\sqrt{-19}) \) is \( \mathbb{Z}[\sqrt{-19}] \), which consists of expressions of _________ ___ _____ __________ ___ __________ ______ _______ _______ _________ ____ ______.
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Let be a finite extension F of odd degree(greater than 1). Show that
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
.
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?
In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group
(a) Let p be an odd prime that divides n, . Suppose
. Show that C is the unique Sylow p-subgroup of Dn .
(b) Prove the relation
Further, find all the elements of order 2 in Dn .
(c) Find all the Sylow 2-subgroups of Dn when n is odd. Describe them in terms of x and y.
(d) Suppose n is even, n = 2km, where . Let
and
. Show that HN is a subgroup of Dn . What is its order?
(e) Suppose n is as in the previous part. Find all the Sylow 2-supgroups of Dn . Describe them in terms of x and y.
Show that, to prove that SP2(R) acts transitively on GL2(R), it is enough to show that, for any vector , there is a
symplectic matrix with
as the first column. (Hint: For any matrix A, what is
?)
Consider the natural action of on
, the set of
real matrices, by left multiplication.
(i) Under this action, if , show that the stabiliser of
is {I}, where I is the 2 × 2 identity matrix.
(ii) Suppose that det(x) = 0 in the remaining parts of this exercise. We will show that the stabiliser of x is infinite. If x = 0, the stabiliser of x is . So suppose x ≠ 0. Let us write
. Then,
for non-zero
. Why ?
(iii) 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.)
(iv) Check that I−b is in the stabiliser of x. Also, show that there are infinitely many choices of for which I − b is invertible.
Compute the following:
a).
b).