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Lattice of subgroups and subfields showing their corresponding Galois groups.
Lattice diagram of adjoin the positive square roots of 2 and 3, its subfields, and Galois groups.

In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. It has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle; a third problem of antiquity, squaring the circle, is also unsolvable, but this is shown by other methods); showing that there is no quintic formula; and showing which polygons are constructible.

The subject is named after Évariste Galois, who introduced it for studying the roots of a polynomial and characterizing the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, nth roots and the four basic arithmetic operations.

The theory has been popularized among mathematicians and developed by Richard Dedekind, Leopold Kronecker, Emil Artin, and others who in particular interpreted the permutation group of the roots as the automorphism group of a field extension.

Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.

Application to classical problemsEdit



Galois' theory not only provides a beautiful answer to this question, but also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do. Further, it gives a conceptually clear, and often practical, means of telling when some particular equation of higher degree can be solved in that manner.

Galois' theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterization of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as

  1. Which regular polygons are constructible polygons?[1]
  2. Why is it not possible to trisect every angle using a compass and straightedge?[1]








A further step was the 1770 paper Réflexions sur la résolution algébrique des équations by the French-Italian mathematician Joseph Louis Lagrange, in his method of Lagrange resolvents, where he analyzed Cardano's and Ferrari's solution of cubics and quartics by considering them in terms of permutations of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois' theory. Crucially, however, he did not consider composition of permutations. Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree.

The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation. His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of the Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel–Ruffini theorem.


Galois' writingsEdit

A portrait of Évariste Galois aged about 15

In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals; Galois' paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois then died in a duel in 1832, and his paper, "Mémoire sur les conditions de résolubilité des équations par radicaux", remained unpublished until 1846 when it was published by Joseph Liouville accompanied by some of his own explanations.[4] Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843.[5] According to Allan Clark, Galois's characterization "dramatically supersedes the work of Abel and Ruffini."[6]


Galois' theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method.[7] Joseph Alfred Serret who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook Cours d'algèbre supérieure. Serret's pupil, Camille Jordan, had an even better understanding reflected in his 1870 book Traité des substitutions et des équations algébriques. Outside France, Galois' theory remained more obscure for a longer period. In Britain, Cayley failed to grasp its depth and popular British algebra textbooks did not even mention Galois' theory until well after the turn of the century. In Germany, Kronecker's writings focused more on Abel's result. Dedekind wrote little about Galois' theory, but lectured on it at Göttingen in 1858, showing a very good understanding.[8] Eugen Netto's books of the 1880s, based on Jordan's Traité, made Galois theory accessible to a wider German and American audience as did Heinrich Martin Weber's 1895 algebra textbook.[9]

Permutation group approach to Galois theoryEdit

Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say A and B, that A2 + 5B3 = 7. The central idea of Galois' theory is to consider permutations (or rearrangements) of the roots such that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. Originally, the theory has been developed for algebraic equations whose coefficients are rational numbers. It extends naturally to equations with coefficients in any field, but this will not be considered in the simple examples below.

These permutations together form a permutation group, also called the Galois group of the polynomial, which is explicitly described in the following examples.

First example: a quadratic equationEdit

Consider the quadratic equation




Examples of algebraic equations satisfied by A and B include




Obviously, in either of these equations, if we exchange A and B, we obtain another true statement. For example, the equation A + B = 4 becomes simply B + A = 4. Furthermore, it is true, but less obvious, that this holds for every possible algebraic relation between A and B such that all coefficients are rational (in any such relation, swapping A and B yields another true relation). This results from the theory of symmetric polynomials, which, in this simple case, may be replaced by formula manipulations involving binomial theorem. (One might object that A and B are related by the algebraic equation AB − 23 = 0, which does not remain true when A and B are exchanged. However, this relation is not considered here, because it has the coefficient −23 which is not rational.)




Second exampleEdit

Consider the polynomial


which can also be written as





Among these equations, we have:


It follows that, if φ is a permutation that belongs to the Galois group, we must have:


This implies that the permutation is well defined by the image of A, and that the Galois group has 4 elements, which are:


This implies that the Galois group is isomorphic to the Klein four-group.

Modern approach by field theoryEdit

In the modern approach, one starts with a field extension L/K (read "L over K"), and examines the group of field automorphisms of L/K (these are bijective ring homomorphisms α : LL such that α(x) = x for all xK). See the article on Galois groups for further explanation and examples.


In the first example above, we were studying the extension Q(3)/Q, where Q is the field of rational numbers, and Q(3) is the field obtained from Q by adjoining 3. In the second example, we were studying the extension Q(A,B,C,D)/Q.



Solvable groups and solution by radicalsEdit



One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals (this was proven independently, using a similar method, by Niels Henrik Abel a few years before, and is the Abel–Ruffini theorem), and a systematic way for testing whether a specific polynomial is solvable by radicals. The Abel–Ruffini theorem results from the fact that for n > 4 the symmetric group Sn contains a simple, noncyclic, normal subgroup, namely the alternating group An.

A non-solvable quintic exampleEdit

For the polynomial f(x) = x5x − 1, the lone real root x = 1.1673… is algebraic, but not expressible in terms of radicals. The other four roots are complex numbers.

Van der Waerden[10] cites the polynomial f(x) = x5x − 1. By the rational root theorem this has no rational zeroes. Neither does it have linear factors modulo 2 or 3.

The Galois group of f(x) modulo 2 is cyclic of order 6, because f(x) modulo 2 factors into polynomials of orders 2 and 3, (x2 + x + 1)(x3 + x2 + 1).


It is known[11] that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. A permutation group on 5 objects with elements of orders 6 and 5 must be the symmetric group S5, which is therefore the Galois group of f(x). This is one of the simplest examples of a non-solvable quintic polynomial. According to Serge Lang, Emil Artin found this example.[12]

Inverse Galois problemEdit

The inverse Galois problem is to find a field extension with a given Galois group

As long as one does not also specify the ground field, the problem is not very difficult, and all finite groups do occur as Galois groups. For showing this, one may proceed as follows. Choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {xα}, one for each element α of G, and adjoin them to K to get the field F = K({xα}). Contained within F is the field L of symmetric rational functions in the {xα}. The Galois group of F/L is S, by a basic result of Emil Artin. G acts on F by restriction of action of S. If the fixed field of this action is M, then, by the fundamental theorem of Galois theory, the Galois group of F/M is G.

On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field Q of the rational numbers. Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of Q. Various people have solved the inverse Galois problem for selected non-Abelian simple groups. Existence of solutions has been shown for all but possibly one (Mathieu group M23) of the 26 sporadic simple groups. There is even a polynomial with integral coefficients whose Galois group is the Monster group.

See alsoEdit


  1. ^ a b Stewart, Ian (1989). Galois Theory. Chapman and Hall. ISBN 0-412-34550-1.
  2. ^ (Funkhouser 1930)
  3. ^ Cardano 1545
  4. ^ Tignol, Jean-Pierre (2001). Galois' Theory of Algebraic Equations. World Scientific. pp. 232–233, 302. ISBN 978-981-02-4541-2.
  5. ^ Stewart, 3rd ed., p. xxiii
  6. ^ Clark, Allan (1984) [1971]. Elements of Abstract Algebra. Courier Corporation. p. 131. ISBN 978-0-486-14035-3.
  8. ^ Scharlau, W., ed. (1981). Richard Dedekind, 1831–1981: Eine Würdigung. Braunschweig, Vieweg.
  9. ^ Galois, Évariste; Neumann, Peter M. (2011). The Mathematical Writings of Évariste Galois. European Mathematical Society. p. 10. ISBN 978-3-03719-104-0.
  12. ^ Lang, Serge (1994), Algebraic Number Theory, Graduate Texts in Mathematics, 110, Springer, p. 121, ISBN 9780387942254.


External linksEdit

Some on-line tutorials on Galois theory appear at:
