# Fundamental theorem of Galois theory

The **fundamental theorem of Galois theory** is a theorem of field theory stating that, for a finite [Galois extension](https://www.edgepedia.org/galois-extension) of fields, the intermediate fields of the extension and the subgroups of its [Galois group](https://www.edgepedia.org/galois-group) are in a bijective, inclusion-reversing correspondence: each intermediate field corresponds to the subgroup of automorphisms that fix it, and each subgroup corresponds to its fixed field.<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup><sup> • </sup><sup>[2](https://jhavaldar.github.io/notes/2017/10/19/galoistheory1.html)</sup> The theorem converts questions about fields, which are hard to compute with directly, into questions about finite groups, and it is the structural core of [Galois theory](https://www.edgepedia.org/galois-theory).<sup>[3](https://notes.amittai.studio/abstract-algebra/galois-theory/the-galois-correspondence)</sup>

| Key fact | Detail |
|---|---|
| Setting | A finite Galois extension \( L/K \), that is, a finite, normal, separable extension, with Galois group \( \mathrm{Gal}(L/K) \)<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup> |
| Correspondence | \( M \mapsto \mathrm{Gal}(L/M) \) is a bijection from intermediate fields \( K \subseteq M \subseteq L \) to subgroups of \( \mathrm{Gal}(L/K) \); its inverse sends a subgroup \( H \) to its fixed field \( L^H \)<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup> |
| Direction | Inclusion reversing: larger fields correspond to smaller subgroups<sup>[2](https://jhavaldar.github.io/notes/2017/10/19/galoistheory1.html)</sup> |
| Degrees | \( [L:M] = |\mathrm{Gal}(L/M)| \) and \( [M:K] = |\mathrm{Gal}(L/K)|/|\mathrm{Gal}(L/M)| \)<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup> |
| Normality | \( M/K \) is Galois if and only if \( \mathrm{Gal}(L/M) \) is a normal subgroup, and then \( \mathrm{Gal}(L/K)/\mathrm{Gal}(L/M) \cong \mathrm{Gal}(M/K) \)<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup> |
| Infinite case | For infinite Galois extensions the bijection holds only between intermediate fields and closed subgroups in the Krull topology, a form due to W. Krull in 1928<sup>[4](https://math.mit.edu/classes/18.785/2021fa/LectureNotes26.pdf)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/topics/mathematics/galois-theory)</sup> |

## Statement

Let \( \iota : K \hookrightarrow L \) be a finite Galois extension, and call a subfield of \( L \) containing \( \iota(K) \) an intermediate field. The theorem asserts three things.<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>

1. The map sending an intermediate field \( M \) to \( G_M = \mathrm{Gal}(L|M) \), the automorphisms of \( L \) fixing \( M \), is a bijection onto the subgroups of \( \mathrm{Gal}(L|K) \). Its inverse sends a subgroup \( H \) to the fixed field \( L^H \), the elements of \( L \) fixed by every element of \( H \).<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>
2. For each intermediate field \( M \), \( [L:M] = |G_M| \) and \( [M:K] = |\mathrm{Gal}(L|K)|/|G_M| \).<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>
3. \( M/K \) is a Galois extension if and only if \( G_M \) is a normal subgroup of \( \mathrm{Gal}(L|K) \); in that case restriction to \( M \) gives an isomorphism \( \mathrm{Gal}(L|K)/G_M \cong \mathrm{Gal}(M|K) \).<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>

The correspondence is inclusion reversing: if \( F \subset K \subset L \subset E \) then \( \{\mathrm{id}\} \subset G(E/L) \subset G(E/K) \subset G(E/F) \).<sup>[6](https://ntouikan.ext.unb.ca/MATH4043/galois-section-fund-theorem-galois-theory.html)</sup> Two more correspondences hold: given fields \( E_1, E_2 \) matched with subgroups \( H_1, H_2 \), the field \( E_1 \cap E_2 \) is matched with the subgroup that \( H_1 \) and \( H_2 \) together generate, while the compositum \( E_1 E_2 \) is matched with the subgroup \( H_1 \cap H_2 \).<sup>[2](https://jhavaldar.github.io/notes/2017/10/19/galoistheory1.html)</sup>

Some texts state the theorem under slightly different hypotheses. The Abstract Algebra: Theory and Applications text assumes \( F \) is a finite field or a field of characteristic zero and \( E \) a finite normal extension of \( F \); a Cambridge course text states it for a finite normal extension inside \( \mathbb{C} \).<sup>[6](https://ntouikan.ext.unb.ca/MATH4043/galois-section-fund-theorem-galois-theory.html)</sup><sup> • </sup><sup>[7](https://math.mit.edu/~dav/galois.pdf)</sup> In characteristic zero every finite extension is separable, so normality alone suffices there.<sup>[8](https://encyclopediaofmath.org/wiki/Galois_theory)</sup>

## Why the correspondence works

Understanding the two directions makes the theorem transparent. One direction is easy: for any subgroup \( H \) of \( \mathrm{Gal}(L/K) \), the fixed field \( L^H \) is a subfield, since elements fixed by automorphisms are closed under addition, subtraction, multiplication, and inverses of nonzero elements.<sup>[6](https://ntouikan.ext.unb.ca/MATH4043/galois-section-fund-theorem-galois-theory.html)</sup> The injectivity of \( H \mapsto L^H \) is the content of Artin's lemma, which the Oxford lecture notes cite as the step showing \( G_{L^H} = H \).<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>

Surjectivity uses that \( L/M \) is itself Galois for every intermediate \( M \). If \( L \) is the splitting field of a separable polynomial \( f(x) \in K[x] \), then \( L \) is also the splitting field of \( f \) over \( M \), so \( L/M \) is Galois and \( M \) is exactly the fixed field of \( \mathrm{Gal}(L/M) \).<sup>[2](https://jhavaldar.github.io/notes/2017/10/19/galoistheory1.html)</sup> The degree identities then follow from the equality of group order and extension degree for a Galois extension, together with the tower law for degrees.<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>

The normality clause has a concrete meaning: an automorphism \( \gamma \) of \( L \) over \( K \) sends \( G_M \) to \( \gamma G_M \gamma^{-1} = \mathrm{Gal}(L|\gamma(M)) \), so \( G_M \) is normal exactly when every \( K \)-automorphism maps \( M \) to itself, which is the condition for \( M/K \) to be normal and separable.<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>

A corollary worth recording: since a finite Galois group has finitely many subgroups, a finite separable extension has only finitely many intermediate fields.<sup>[1](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)</sup>

## Worked example: the splitting field of x⁴ − 2

The splitting field of \( f(x) = x^4 - 2 \) over \( \mathbb{Q} \) is \( \mathbb{Q}(\sqrt[4]{2}, i) \), obtained by adjoining \( \sqrt[4]{2} \) and then \( i \). The Galois group has order 8; with \( \sigma(\sqrt[4]{2}) = i\sqrt[4]{2} \), \( \sigma(i) = i \), and \( \tau \) complex conjugation, the relations \( \tau^2 = \mathrm{id} \), \( \sigma^4 = \mathrm{id} \), and \( \tau\sigma\tau = \sigma^{-1} \) hold, so the group is isomorphic to the dihedral group \( D_4 \).<sup>[6](https://ntouikan.ext.unb.ca/MATH4043/galois-section-fund-theorem-galois-theory.html)</sup> The theorem says the lattice of subgroups of \( D_4 \) matches, in reverse order, the lattice of intermediate fields between \( \mathbb{Q} \) and \( \mathbb{Q}(\sqrt[4]{2}, i) \), so listing subgroups, a finite group-theoretic task, enumerates all intermediate fields.<sup>[6](https://ntouikan.ext.unb.ca/MATH4043/galois-section-fund-theorem-galois-theory.html)</sup><sup> • </sup><sup>[9](https://calc-path.com/abstract-algebra/modules/galois-theory)</sup>

## Finite fields

Finite fields give the correspondence in its simplest form. The extension \( \mathbb{F}_{p^n}/\mathbb{F}_p \) is Galois of degree \( n \) with cyclic Galois group \( \mathbb{Z}/n\mathbb{Z} \) generated by the Frobenius endomorphism \( F(\alpha) = \alpha^p \). Because the group is abelian, every subgroup is normal, so by the fundamental theorem every subextension of \( \mathbb{F}_{p^n}/\mathbb{F}_p \) is Galois over \( \mathbb{F}_p \). Each divisor \( d \) of \( n \) yields precisely one subfield of degree \( d \) over \( \mathbb{F}_p \), namely the elements left fixed by the group that \( F^{n/d} \) generates, and this subfield is isomorphic to \( \mathbb{F}_{p^d} \).<sup>[10](https://web.ma.utexas.edu/users/ikmartin/notes/finite-fields-existence-and-galois-theory/)</sup>

## The infinite case

For infinite Galois extensions the naive correspondence fails because there are too many subgroups. The absolute Galois group of \( \mathbb{Q} \) has uncountably many subgroups of index 2, while \( \mathbb{Q} \) has only countably many quadratic extensions; subgroups with the same closure have the same fixed field.<sup>[4](https://math.mit.edu/classes/18.785/2021fa/LectureNotes26.pdf)</sup> W. Krull resolved this in 1928 using the profinite topology now called the Krull topology, whose basic open sets are cosets of subgroups \( \mathrm{Gal}(L/F) \) for finite subextensions \( F/K \).<sup>[4](https://math.mit.edu/classes/18.785/2021fa/LectureNotes26.pdf)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/topics/mathematics/galois-theory)</sup> With this topology, the maps \( F \mapsto \mathrm{Gal}(L/F) \) and \( H \mapsto L^H \) give an inclusion-reversing bijection between subextensions of \( L/K \) and closed subgroups of \( \mathrm{Gal}(L/K) \); subextensions of finite degree \( n \) correspond to closed subgroups of finite index \( n \), and normal subextensions correspond to normal closed subgroups \( H \) with \( \mathrm{Gal}(F/K) \cong G/H \) as topological groups.<sup>[4](https://math.mit.edu/classes/18.785/2021fa/LectureNotes26.pdf)</sup> Under this correspondence the Galois group is a profinite group, compact and totally disconnected.<sup>[4](https://math.mit.edu/classes/18.785/2021fa/LectureNotes26.pdf)</sup>

## History

[Évariste Galois](https://www.edgepedia.org/evariste-galois) (1811–1832) developed the ideas underlying the theorem in work on the solvability of equations by radicals; he defined the group of an equation as the permutations of the roots that preserve all rationally expressible relations among them, and introduced the notion of what is now called a normal subgroup.<sup>[11](https://mathshistory.st-andrews.ac.uk//Projects/Brunk/chapter-3/)</sup><sup> • </sup><sup>[12](https://mathshistory.st-andrews.ac.uk/Extras/Burnside_groups/)</sup> His main writings were published in French in 1846 by Joseph Liouville, 14 years after Galois died in a duel at age 20.<sup>[13](https://ems.press/books/hem/102)</sup> Galois derived the deeper structure from group theory after defining normal subgroups, and later commentators such as Dedekind and Weber developed the field-theoretic foundations.<sup>[5](https://www.sciencedirect.com/topics/mathematics/galois-theory)</sup><sup> • </sup><sup>[14](https://mathshistory.st-andrews.ac.uk/Projects/Brunk/chapter-5/)</sup> [Emil Artin](https://www.edgepedia.org/emil-artin) established Galois theory as the study of fields and their automorphism groups, combining earlier approaches into one coherent theory; from that point the theorem's modern form, built on fixed fields and automorphism groups, became standard.<sup>[14](https://mathshistory.st-andrews.ac.uk/Projects/Brunk/chapter-5/)</sup>

## Applications and later work

The theorem is the engine behind the classical applications of Galois theory. It underlies the criterion that a polynomial is solvable by radicals when its Galois group is solvable, a group-theoretic property defined by a subnormal series with abelian factor groups; \( S_3 \) is solvable while \( S_5 \) is not.<sup>[15](https://ntouikan.ext.unb.ca/MATH4043/galois-applications-section-solvable-by-radicals.html)</sup> It also yields Gauss's theorem on constructibility of regular polygons with straightedge and compass.<sup>[16](https://www.math.ucdavis.edu/~egorskiy/MAT150C-s21/Galois.pdf)</sup> In the finite-field setting the correspondence drives the subfield structure used in applications such as cryptographic algorithms, including the [Advanced Encryption Standard](https://www.edgepedia.org/advanced-encryption-standard), that rely on arithmetic in extensions of finite fields.<sup>[17](https://www.mi.uni-koeln.de/~mmertens/GaloisTheory.pdf)</sup>

Proof technique remains a live topic: a 2022 arXiv paper gives a simplified proof of the finite theorem that relies on the combinatorial fact that a field cannot be written as a union of finitely many proper subfields.<sup>[18](https://arxiv.org/abs/2211.07508)</sup>

## References

1. [The fundamental theorem of Galois theory, Section 4.3, lecture notes of D. Rössler, University of Oxford](https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/GTnotesDR.pdf)
2. [Jay Havaldar, Galois Theory, Part 1: The Fundamental Theorem of Galois Theory](https://jhavaldar.github.io/notes/2017/10/19/galoistheory1.html)
3. [The Galois Correspondence, Amittai study notes on abstract algebra](https://notes.amittai.studio/abstract-algebra/galois-theory/the-galois-correspondence)
4. [18.785 Number Theory I, Lecture 26: Infinite Galois theory, MIT (Fall 2021)](https://math.mit.edu/classes/18.785/2021fa/LectureNotes26.pdf)
5. [Galois theory, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Galois_theory)
6. [AATA, Section 8.2: The Fundamental Theorem](https://ntouikan.ext.unb.ca/MATH4043/galois-section-fund-theorem-galois-theory.html)
7. [18.704 Seminar in Algebra and Number Theory, MIT: The Fundamental Theorem](https://math.mit.edu/~dav/galois.pdf)
8. [Galois theory overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/galois-theory)
9. [CalcPath, Abstract Algebra: Galois theory, Section 8.4](https://calc-path.com/abstract-algebra/modules/galois-theory)
10. [Isaac Martin, Finite Fields: Existence and Galois Theory, University of Texas notes](https://web.ma.utexas.edu/users/ikmartin/notes/finite-fields-existence-and-galois-theory/)
11. [Fiona Brunk, The development of Galois theory, Chapter 3: Évariste Galois, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Projects/Brunk/chapter-3/)
12. [W. Burnside, Theory of Groups of Finite Order, preface reproduced at MacTutor](https://mathshistory.st-andrews.ac.uk/Extras/Burnside_groups/)
13. [The mathematical writings of Évariste Galois, EMS Press](https://ems.press/books/hem/102)
14. [Fiona Brunk, The development of Galois theory, Summary, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Projects/Brunk/chapter-5/)
15. [AATA, Section 11.1: Solvability by Radicals](https://ntouikan.ext.unb.ca/MATH4043/galois-applications-section-solvable-by-radicals.html)
16. [Lecture notes on Galois theory, UC Davis MAT 150C](https://www.math.ucdavis.edu/~egorskiy/MAT150C-s21/Galois.pdf)
17. [M. Mertens, Galois Theory lecture notes, University of Cologne](https://www.mi.uni-koeln.de/~mmertens/GaloisTheory.pdf)
18. [A simple proof of the fundamental theorem of Galois theory, arXiv:2211.07508](https://arxiv.org/abs/2211.07508)

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