Galois theory converts field extensions into groups of symmetries. The fundamental theorem says that intermediate fields and subgroups determine each other in reverse order.
Automorphisms fixing the base field
Definition. For a field extension \(L/K\), the Galois group is \[ \operatorname{Gal}(L/K)=\{\sigma\in\operatorname{Aut}(L):\sigma(a)=a\text{ for all }a\in K\}. \]
Example. \(\operatorname{Gal}(\mathbb C/\mathbb R)=\{1,\text{complex conjugation}\}\cong C_2\). The automorphism is determined by where it sends \(i\), and it must send \(i\) to another root of \(x^2+1\).
Fixed fields
Definition. If \(H\le\operatorname{Gal}(L/K)\), the fixed field of \(H\) is \[ L^H=\{x\in L:\sigma(x)=x\text{ for all }\sigma\in H\}. \]
Proposition. Larger subgroups have smaller fixed fields: if \(H_1\subset H_2\), then \(L^{H_2}\subset L^{H_1}\).
Fundamental theorem of Galois theory
Theorem. If \(L/K\) is finite Galois with group \(G\), then there is an inclusion-reversing bijection \[ \{E:K\subset E\subset L\} \longleftrightarrow \{H:H\le G\}, \qquad E\mapsto\operatorname{Gal}(L/E),\quad H\mapsto L^H. \]
Theorem. Under this correspondence, \(E/K\) is Galois iff \(\operatorname{Gal}(L/E)\trianglelefteq G\). In that case \[ \operatorname{Gal}(E/K)\cong G/\operatorname{Gal}(L/E). \]
Proof. The maps \(E\mapsto\operatorname{Gal}(L/E)\) and \(H\mapsto L^H\) reverse inclusions by definition. For finite Galois \(L/K\), Artin's fixed-field theorem gives \([L:L^H]=|H|\), while separability gives \([L:E]=|\operatorname{Gal}(L/E)|\). These degree equalities force the two constructions to be inverse. Normality of \(\operatorname{Gal}(L/E)\) is exactly the condition that automorphisms of \(L/K\) descend to automorphisms of \(E/K\).
Example: a biquadratic extension
Example. Let \(L=\mathbb Q(\sqrt2,\sqrt3)\). The Galois group is \(C_2\times C_2\), generated by independently changing signs of \(\sqrt2\) and \(\sqrt3\). The three subgroups of order \(2\) correspond to the three quadratic fields \(\mathbb Q(\sqrt2)\), \(\mathbb Q(\sqrt3)\), and \(\mathbb Q(\sqrt6)\).
Example: x cubed minus two
Calculation. The splitting field of \(x^3-2\) is \(L=\mathbb Q(\sqrt[3]{2},\zeta_3)\). It has degree \(6\), and its Galois group is isomorphic to \(S_3\). One automorphism cycles the three roots, and another swaps the two nonreal roots by complex conjugation.
Solvability by radicals
Remark. Galois theory explains why polynomial equations connect to group theory. A polynomial is solvable by radicals only when its Galois group has the appropriate solvability property. The generic quintic has Galois group \(S_5\), which is not solvable.
Fixed fields and degree inequalities
Artin's lemma. Distinct field homomorphisms \(K\to L\) are linearly independent as functions.
Proof. Suppose a shortest nontrivial relation \(\sum_{i=1}^na_i\sigma_i=0\) exists. Choose \(x\) with \(\sigma_1(x)\ne\sigma_n(x)\). Evaluate at \(xy\) and subtract \(\sigma_n(x)\) times the relation evaluated at \(y\). The \(\sigma_n\) term vanishes and the \(\sigma_1\) coefficient remains nonzero, contradicting minimality.
Theorem (Artin). If a finite group \(G\) acts faithfully by automorphisms on a field \(L\), then \[ [L:L^G]=|G|,\qquad \operatorname{Gal}(L/L^G)=G. \]
Linear independence gives \([L:L^G]\ge|G|\). Conversely, orbit polynomials \[ \prod_{\sigma\in G}(t-\sigma(\alpha)) \] have coefficients in \(L^G\), so every \(\alpha\) has degree at most \(|G|\); a finite generating argument gives the reverse inequality.
Fundamental correspondence
Fundamental theorem of Galois theory. For finite Galois \(L/K\), the maps \[ H\longmapsto L^H,\qquad E\longmapsto\operatorname{Gal}(L/E) \] are inverse inclusion-reversing bijections between subgroups of \(\operatorname{Gal}(L/K)\) and intermediate fields. Moreover, \[ [L:E]=|\operatorname{Gal}(L/E)|,\qquad [E:K]=[G:H]. \] The extension \(E/K\) is Galois exactly when \(H\triangleleft G\), in which case \(\operatorname{Gal}(E/K)\cong G/H\).
Proof. Artin's theorem gives \([L:L^H]=|H|\). For an intermediate field \(E\), separability and normality of \(L/K\) imply \(|\operatorname{Gal}(L/E)|=[L:E]\), so degree comparison forces both compositions to be identities. An automorphism \(\sigma\in G\) sends \(L^H\) to \(L^{\sigma H\sigma^{-1}}\); hence \(L^H\) is stable under \(G\) exactly when \(H\) is normal. Restriction then has kernel \(H\) and image \(\operatorname{Gal}(E/K)\).