Abstract Algebra: Theory and Applications

Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of $\mathbb{Q}$ in which the extension field is contained.

Give the order and describe a generator of the Galois group of $\mathrm{GF}(729)$ over $\mathrm{GF}(9)\text{.}$

Prove that the Galois group of an irreducible quadratic polynomial is isomorphic to ${\mathbb{Z}}_2\text{.}$

Prove that the Galois group of an irreducible cubic polynomial is isomorphic to $S_3$ or ${\mathbb{Z}}_3\text{.}$

Let $F\subset K\subset E$ be fields. If $E$ is a normal extension of $F\text{,}$ show that $E$ must also be a normal extension of $K\text{.}$

Let $G$ be the Galois group of a polynomial of degree $n\text{.}$ Prove that $|G|$ divides $n!\text{.}$

Let $F\subset E\text{.}$ If $f(x)$ is solvable over $F\text{,}$ show that $f(x)$ is also solvable over $E\text{.}$

Construct a polynomial $f(x)$ in $\mathbb{Q}[x]$ of degree $7$ that is not solvable by radicals.

Let $p$ be prime. Prove that there exists a polynomial $f(x)\in \mathbb{Q}[x]$ of degree $p$ with Galois group isomorphic to $S_p\text{.}$ Conclude that for each prime $p$ with $p\ge 5$ there exists a polynomial of degree $p$ that is not solvable by radicals.

Let $p$ be a prime and ${\mathbb{Z}}_p(t)$ be the field of rational functions over ${\mathbb{Z}}_p\text{.}$ Prove that $f(x)=x^p-t$ is an irreducible polynomial in ${\mathbb{Z}}_p(t)[x]\text{.}$ Show that $f(x)$ is not separable.

Let $E$ be an extension field of $F\text{.}$ Suppose that $K$ and $L$ are two intermediate fields. If there exists an element $\sigma \in G(E/F)$ such that $\sigma (K)=L\text{,}$ then $K$ and $L$ are said to be conjugate fields. Prove that $K$ and $L$ are conjugate if and only if $G(E/K)$ and $G(E/L)$ are conjugate subgroups of $G(E/F)\text{.}$

Let $\sigma \in \mathrm{Aut}(\mathbb{R})\text{.}$ If $a$ is a positive real number, show that $\sigma (a)>0\text{.}$

Let $K$ be the splitting field of $x^3+x^2+1\in {\mathbb{Z}}_2[x]\text{.}$ Prove or disprove that $K$ is an extension by radicals.

Let $F$ be a field such that $\mathrm{char}(F)\ne 2\text{.}$ Prove that the splitting field of $f(x)=a{x}^2+bx+c$ is $F(\sqrt{\alpha })\text{,}$ where $\alpha =b^2-4ac\text{.}$

Prove or disprove: Two different subgroups of a Galois group will have different fixed fields.

Let $K$ be the splitting field of a polynomial over $F\text{.}$ If $E$ is a field extension of $F$ contained in $K$ and $[E:F]=2\text{,}$ then $E$ is the splitting field of some polynomial in $F[x]\text{.}$

Let $F$ be a finite field or a field of characteristic zero. Let $E$ be a finite normal extension of $F$ with Galois group $G(E/F)\text{.}$ Prove that $F\subset K\subset L\subset E$ if and only if $\{\mathrm{id}\}\subset G(E/L)\subset G(E/K)\subset G(E/F)\text{.}$