# Galois Groups ## Symmetry of Polynomial Roots A polynomial equation may possess several roots related by hidden algebraic symmetries. Consider $$ f(x)=x^2-2. $$ Its roots are $$ \sqrt{2} \qquad \text{and} \qquad -\sqrt{2}. $$ These roots can be exchanged without affecting any rational number. Indeed, the transformation $$ \sqrt{2}\mapsto -\sqrt{2} $$ preserves all algebraic relations involving rational coefficients. For example, $$ (\sqrt{2})^2=2 $$ remains true after the substitution. Such transformations are called automorphisms of fields. The set of all automorphisms preserving the base field forms the Galois group. The fundamental idea of Galois theory is that the arithmetic structure of a polynomial is encoded in the symmetries of its roots. ## Field Automorphisms Let $L$ be a field. **Definition.** A field automorphism of $L$ is a bijection $$ \sigma:L\to L $$ satisfying $$ \sigma(a+b)=\sigma(a)+\sigma(b) $$ and $$ \sigma(ab)=\sigma(a)\sigma(b) $$ for all $a,b\in L$. Since multiplication is preserved, inverses are also preserved: $$ \sigma(a^{-1})=\sigma(a)^{-1} $$ whenever $a\neq 0$. An automorphism therefore respects the entire field structure. If $K\subseteq L$, we are usually interested in automorphisms that leave every element of $K$ fixed. **Definition.** The Galois group of the extension $L/K$ is $$ \mathrm{Gal}(L/K), $$ the group of all field automorphisms of $L$ that fix $K$ pointwise. Thus, $$ \sigma(a)=a \qquad \text{for all } a\in K. $$ Composition of maps gives the group operation. ## First Example Consider the extension $$ \mathbb{Q}(\sqrt{2})/\mathbb{Q}. $$ Every element has the form $$ a+b\sqrt{2}, \qquad a,b\in\mathbb{Q}. $$ An automorphism fixing $\mathbb{Q}$ must send $\sqrt{2}$ to another root of its minimal polynomial $$ x^2-2. $$ The roots are $$ \sqrt{2} \qquad \text{and} \qquad -\sqrt{2}. $$ Therefore there are exactly two automorphisms: 1. The identity automorphism $$ \sigma(a+b\sqrt{2})=a+b\sqrt{2}. $$ 2. The conjugation automorphism $$ \tau(a+b\sqrt{2})=a-b\sqrt{2}. $$ Hence $$ \mathrm{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q}) $$ contains two elements. As a group, it is isomorphic to the cyclic group $$ C_2. $$ The extension degree is also $2$: $$ [\mathbb{Q}(\sqrt{2}):\mathbb{Q}]=2. $$ This equality between extension degree and group size is a central feature of Galois extensions. ## Automorphisms and Polynomial Roots Let $L$ be the splitting field of a polynomial $$ f(x)\in K[x]. $$ Every automorphism in $$ \mathrm{Gal}(L/K) $$ permutes the roots of $f(x)$. Indeed, suppose $\alpha$ is a root: $$ f(\alpha)=0. $$ Applying an automorphism $\sigma$, we obtain $$ \sigma(f(\alpha))=\sigma(0)=0. $$ Since $\sigma$ fixes the coefficients of $f(x)$, $$ f(\sigma(\alpha))=0. $$ Thus $\sigma(\alpha)$ is again a root of the polynomial. Consequently, the Galois group acts as a permutation group on the roots. This observation connects field theory with group theory. ## Example: $x^3-2$ Consider $$ f(x)=x^3-2 $$ over $\mathbb{Q}$. Its splitting field is $$ L=\mathbb{Q}(\sqrt[3]{2},\omega), $$ where $\omega$ is a primitive cube root of unity. The roots are $$ \sqrt[3]{2}, \qquad \omega\sqrt[3]{2}, \qquad \omega^2\sqrt[3]{2}. $$ Any automorphism fixing $\mathbb{Q}$ must permute these roots. One possible automorphism sends $$ \sqrt[3]{2}\mapsto \omega\sqrt[3]{2} $$ while fixing $\omega$. Another sends $$ \omega\mapsto \omega^2. $$ Together these generate a group of order $6$, isomorphic to the symmetric group $$ S_3. $$ Thus $$ \mathrm{Gal}(L/\mathbb{Q})\cong S_3. $$ The polynomial $x^3-2$ therefore possesses the full symmetry structure of permutations on three objects. ## Fixed Fields Given a subgroup $$ G\subseteq \mathrm{Gal}(L/K), $$ one may consider the elements fixed by every automorphism in $G$. **Definition.** The fixed field of $G$ is $$ L^G= \{ a\in L:\sigma(a)=a \text{ for all } \sigma\in G \}. $$ For example, in $$ \mathbb{Q}(\sqrt{2})/\mathbb{Q}, $$ the nontrivial automorphism sends $$ a+b\sqrt{2}\mapsto a-b\sqrt{2}. $$ The only elements unchanged are those with $b=0$. Hence the fixed field is precisely $$ \mathbb{Q}. $$ The interaction between subgroups and fixed fields leads to the fundamental theorem of Galois theory. ## Galois Extensions Not every field extension behaves perfectly with respect to automorphisms. The best situation occurs when the extension is both normal and separable. **Definition.** A finite extension $L/K$ is called a Galois extension if 1. every irreducible polynomial in $K[x]$ having one root in $L$ splits completely in $L$; 2. every irreducible polynomial has distinct roots. For fields of characteristic zero, separability is automatic. Thus finite splitting fields over $\mathbb{Q}$ are Galois extensions. If $L/K$ is Galois, then $$ |\mathrm{Gal}(L/K)|=[L:K]. $$ This equality reveals that the algebraic size of the extension equals the number of its symmetries. ## The Fundamental Theorem of Galois Theory The central theorem of Galois theory establishes a correspondence between fields and groups. Let $L/K$ be a finite Galois extension. Then: - every intermediate field $$ K\subseteq E\subseteq L $$ corresponds to a subgroup $$ \mathrm{Gal}(L/E); $$ - every subgroup $$ H\subseteq \mathrm{Gal}(L/K) $$ corresponds to a fixed field $$ L^H. $$ These correspondences reverse inclusion: $$ E_1\subseteq E_2 \quad\Longleftrightarrow\quad \mathrm{Gal}(L/E_2)\subseteq \mathrm{Gal}(L/E_1). $$ Thus algebraic information about field extensions becomes equivalent to group-theoretic information about automorphisms. This theorem is one of the deepest structural results in mathematics. ## Solvability by Radicals The historical origin of Galois theory was the study of polynomial equations. Quadratic, cubic, and quartic equations admit formulas involving radicals. However, general quintic equations do not. The reason lies in group theory. A polynomial is solvable by radicals precisely when its Galois group is a solvable group. For example: - quadratic equations correspond to groups of order $2$; - general cubic equations correspond to subgroups of $S_3$; - general quintic equations correspond to $S_5$, which is not solvable. Thus the impossibility of a general quintic formula is ultimately a theorem about symmetry. ## Galois Groups in Number Theory Galois groups govern much of modern number theory. Cyclotomic fields, finite fields, elliptic curves, modular forms, and algebraic varieties all possess associated Galois actions. Arithmetic information is frequently encoded in how the absolute Galois group $$ \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) $$ acts on algebraic objects. Modern theories such as class field theory and the Langlands program may be viewed as vast generalizations of the correspondence between arithmetic and symmetry first discovered by Évariste Galois.