# 940 Extra Problems Galois Theory

## Galois Theory

### Theory

• Show that if $$K/F$$ is the splitting field of a separable polynomial then it is Galois.
• Show that any quadratic extension of a field $$F$$ with $$\operatorname{ch}(F)\neq 2$$ is Galois.
• Show that if $$K/E/F$$ with $$K/F$$ Galois then $$K/E$$ is always Galois with $$g(K/E) \leq g(K/F)$$.
• Show additionally $$E/F$$ is Galois $$\iff g(K/E) {~\trianglelefteq~}g(K/F)$$.
• Show that in this case, $$g(E/F) = g(K/F) / g(K/E)$$.
• Show that if $$E/k, F/k$$ are Galois with $$E\cap F = k$$, then $$EF/k$$ is Galois and $$G(EF/k) \cong G(E/k)\times G(F/k)$$.

### Computations

• Show that the Galois group of $$x^n - 2$$ is $$D_n$$, the dihedral group on $$n$$ vertices.
• Compute all intermediate field extensions of $${\mathbf{Q}}(\sqrt 2, \sqrt 3)$$, show it is equal to $${\mathbf{Q}}(\sqrt 2 + \sqrt 3)$$, and find a corresponding minimal polynomial.

• Compute all intermediate field extensions of $${\mathbf{Q}}(2^{1\over 4}, \zeta_8)$$.
• Show that $${\mathbf{Q}}(2^{1\over 3})$$ and $${\mathbf{Q}}(\zeta_3 2^{1\over 3})$$
• Show that if $$L/K$$ is separable, then $$L$$ is normal $$\iff$$ there exists a polynomial $$p(x) = \prod_{i=1}^n x- \alpha_i\in K[x]$$ such that $$L = K(\alpha_1, \cdots, \alpha_n)$$ (so $$L$$ is the splitting field of $$p$$).
• Is $${\mathbf{Q}}(2^{1\over 3})/{\mathbf{Q}}$$ normal?
• Show that $${\mathbf{GF}}(p^n)$$ is the splitting field of $$x^{p^n} - x \in { \mathbf{F} }_p[x]$$.
• Show that $${\mathbf{GF}}(p^d) \leq {\mathbf{GF}}(p^n) \iff d\divides n$$
• Compute the Galois group of $$x^n - 1 \in {\mathbf{Q}}[x]$$ as a function of $$n$$.
• Identify all of the elements of the Galois group of $$x^p-2$$ for $$p$$ an odd prime (note: this has a complicated presentation).
• Show that $${ \operatorname{Gal}}(x^{15}+2)/{\mathbf{Q}}\cong S_2 \rtimes{\mathbf{Z}}/15{\mathbf{Z}}$$ for $$S_2$$ a Sylow $$2{\hbox{-}}$$subgroup.
• Show that $${ \operatorname{Gal}}(x^3+4x+2)/{\mathbf{Q}}\cong S_3$$, a symmetric group.