Find where ?3 = 1, ? ≠ 1.
To solve the given problem, we need to compute the degree of the field extension \(\mathbb{Q}(\sqrt[3]{2}, \omega)\) over \(\mathbb{Q}\), where \(\omega\) is a cube root of unity (i.e., \(\omega^3 = 1\), and \(\omega \neq 1\)). ### Step 1: Understanding the field \(\mathbb{Q}(\sqrt[3]{2})\) The element \(\sqrt[3]{2}\) ____ _______ _____ __________ ____ _________.
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Find where ?3 = 1, ? ≠ 1.
Consider the natural action of GL2 on M2
, the set of 2 x 2 real matrices, by left multiplication.
(i) Under this action, if det(x) ≠ 0, show that the stabiliser of x ∈ M2 is {I}, where I is the 2 x 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 GL2. So suppose x ≠ 0. Let us write
Then,
for non-zero λ ∈ R. Why?
(iii) Let be a vector that is not a scalar multiple of
. Show that there is a matrix b =
such that b
= 0 and b
= α
(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 a for which I - b is invertible.
Let p be an odd prime that divides n, n = pr l, p + 1. Suppose C = (x1). Show that C is the unique Sylow p-subgroup of Dn.
Show that
(i) Show that a matrix is symplectic if and only if ad - bc = 1.
(ii) Show that, to prove that SP2 acts transitively on GL2
, it is enough to show that, for any vector
, there is a 2 x
Find the elementary divisors and invariant factors of
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 . Show that G is the cyclic group of order six.