Generic Math Blog

Prove that \(\mathbb{F}_p(x,y)/\mathbb{F}_p(x^p,y^p)\) is not a simple extension by explicitly exhibiting an infinite number of infinite sub...

Show that \(\mathbb{Q}(\sqrt{2+\sqrt{2}})\) is a cyclic quartic field. Let \(a_+ = \sqrt{2+\sqrt{2}}\) and \(a_- = \sqrt{2-\sqrt{2}}\). It c...

Determine the Galois group of the splitting field over \(\mathbb{Q}\) of \(x^4-14x^2+9\). We can solve for the roots explicitly: they are \(...

Suppose \(f(x) \in \mathbb{Z}[x]\) is an irreducible quartic whose splitting field has Galois group \(S_4\) over \(\mathbb{Q}\). Let \(\thet...

Prove that the automorphisms of the rational function field \(k(t)\) which fix \(k\) are precisely the fractional linear transformations det...

We start by mentioning the following result: If a field extension \(K/F\) has degree \(n\), then any \(\alpha \in K\) satisfies a polynomial...

Suppose \(m\) and \(n\) are relatively prime positive integers. Let \(\zeta_m\) and \(\zeta_n\) be primitive \(m\)th and \(n\)th roots of un...