Galois Theory

This is a reference document on Galois Theory by Akshay Tiwary and Fengyang Wang. It covers a subset of the material of PMATH 348.


A field is a collection of objects which can be added and subtracted, and further can be multiplied and (except zero) divided. Furthermore, the multiplication distributes over addition. For example: $\mathbf{Q}$, the rational numbers; $\mathbf{R}$, the real numbers; $\mathbf{C}$, the complex numbers.

When a field is contained within another field, e.g. $\mathbf{Q}\subseteq\mathbf{C}$, we call the larger field a \textbf{field extension} of the smaller field. Note that the larger field is naturally a vector space over the smaller field, by only allowing multiplication of elements of the larger field by elements of the smaller field.

If $K \subseteq L$ are fields, then an element $\alpha\in L$ is algebraic over $K$ if it is the root of some nonzero polynomial with coefficients in $K$. For instance $i$ is a root of $x^2 + 1$ so $i$ is algebraic over $\mathbf{Q}$. $\pi$ is transcendental (not algebraic) over $\mathbf{Q}$.

If $L$ contains only algebraic elements over $K$, then $L$ is an algebraic extension of $K$. Otherwise $L$ is a transcendental extension. For instance $\mathbf{Q} \subseteq \mathbf{C}$ is transcendental, whereas $\mathbf{R} \subseteq \mathbf{C}$ is algebraic. Can you see why?

If $a$ is algebraic over $K$ then the minimal polynomial of $a$ over $K$ is the smallest degree nonzero monic polynomial with coefficients in K with $a$ as a root.

Let $K \subseteq L$ be an extension. Then $[L : K] = \operatorname{dim}_K L$ is the degree of $L$ over $K$. We will use that $[K(a) : K] = \operatorname{deg}_K a$.

If $K \subseteq L$ are fields, and $S \subseteq L$ is a set, then $K(S)$ is the smallest subfield of $L$ (i.e. $K(S)\subseteq L$) such that $S \subseteq K(S)$ and $K \subseteq K(S)$. For example

\[\mathbf{Q}(\sqrt[3]{2}) = \{a + b\sqrt[3]{2} + c\sqrt[3]{2}^2 : (a, b, c) \in \mathbf{Q}^3\}\]

(Exercise: Check that this is a field, and that it's smallest.)

Galois Theory

From now on, we mostly work with $\mathbf{Q}\subseteq\mathbf{C}$.

If $p$ is a nonzero polynomial with coefficients in $\mathbf{Q}$, then the splitting field of $p$ is the smallest field containing all roots of $p$. For example the splitting field of $x^2 + 1$ is

\[\mathbf{Q}(i) = \{a + bi : (a, b) \in \mathbf{Q}^2\}\]

Let $K \subseteq L$. An automorphism $\varphi$ of the extension $L/K$ is a map that preserves the field structure and the extension structure. (That is, it fixes all elements of $K$, and is compatible with the field operations.) More concretely $\varphi: L \to L$ is a bijection such that $\varphi(ab^{-1}-c) = \varphi(a)\varphi(b)^{-1} - \varphi(c)$ for $(a, b, c) \in L^3$ and $\varphi(k) = k$ for $k\in K$.

If $\mathbf{Q} \subseteq K \subseteq L \subseteq \mathbf{C}$ with $L$ finite-dimensional over $K$ as a vector space, and $L$ is the splitting field for some $f(x)\in K[x]$, then we say $L/K$ is Galois. Furthermore we define $\operatorname{Gal}(L/K)$ the Galois group of $L/K$ to be the set of all automorphisms of $L/K$.

If $\mathbf{Q} \subseteq K \subseteq L \subseteq \mathbf{C}$ with $L/K$ Galois, then the fundamental theorem of Galois theory says:

  • there is a correspondence between subfields of $L$ that contain $K$ and the subgroups of $\operatorname{Gal}(L/K)$

  • the normal subgroups of $\operatorname{Gal}(L/K)$ correspond to Galois extensions of $K$

  • $|\operatorname{Gal}(L/K)| = [K : L]$