In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, 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. Euclid, detail from The School of Athens by Raphael. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Galois at the age of fifteen from the pencil of a classmate. ... Field theory is a branch of mathematics which studies the properties of fields. ... Group theory is that branch of mathematics concerned with the study of groups. ...

Originally Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions. In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In abstract algebra, a subfield of a field L is a subset K of L which is closed under the addition and multiplication operations of L and itself forms a field with these operations. ...

Further abstraction of Galois theory is achieved by the theory of Galois connections. In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ...

Application to classical problems

The birth of Galois theory was originally motivated by the following question, which is known as the Abel-Ruffini theorem. The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ...

"Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?"

Galois theory not only provides a beautiful answer to this question, it 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.

Galois theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterisation 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 Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ...

"Which regular polygons are constructible polygons?"

"Why is it not possible to trisect every angle?"

Look up Polygon in Wiktionary, the free dictionary. ... In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ...

The permutation group approach to Galois theory

If we are given a polynomial, it may happen that some of the roots of the polynomial are connected by various algebraic equations. For example, it may turn out that for two of the roots, say A and B, the equation A² + 5B³ = 7 holds. The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers. (One might instead specify a certain field in which the coefficients should lie, but for the simple examples below, we will restrict ourselves to the field of rational numbers.) In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... This article presents the essential definitions. ...

These permutations together form a permutation group, also called the Galois group of the polynomial (over the rational numbers). This can be made much clearer by way of example. In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

First example — a quadratic equation

Consider the quadratic equation Graph of a quadratic function: y = x2−x−2 = (x+1)(x−2) The x-coordinates of the points where the graph crosses the x-axis, x = −1 and x = 2, are the roots of the quadratic equation: x2−x−2 = 0 In mathematics, a quadratic equation is a polynomial...

x² − 4x + 1 = 0.

By using the quadratic formula, we find that the two roots are Graph of a quadratic function: y = x2−x−2 = (x+1)(x−2) The x-coordinates of the points where the graph crosses the x-axis, x = −1 and x = 2, are the roots of the quadratic equation: x2−x−2 = 0 In mathematics, a quadratic equation is a polynomial...

A = 2 + √3,   and

B = 2 − √3.

Examples of algebraic equations satisfied by A and B include

A + B = 4,   and

AB = 1.

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 far less obvious, that this holds for every possible algebraic equation satisfied by A and B; to prove this requires the theory of symmetric polynomials. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ...

We conclude that the Galois group of the polynomial x² − 4x + 1 consists of two permutations: the identity permutation which leaves A and B untouched, and the transposition permutation which exchanges A and B. As a group, it is isomorphic to the cyclic group of order two, denoted Z/2Z. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

One might raise the objection that A and B are related by yet another algebraic equation,

A − B − 2√3 = 0,

which does not remain true when A and B are exchanged. However, this equation does not concern us, because it does not have rational coefficients; in particular, √3 is not rational. In mathematics, an irrational number is any real number that is not a rational number, i. ...

A similar discussion applies to any quadratic polynomial ax² + bx + c, where a, b and c are rational numbers.

• If the polynomial has only one root, for example x² − 4x + 4 = (x−2)², then the Galois group is trivial; that is, it contains only the identity permutation.
• If it has two distinct rational roots, for example x² − 3x + 2 = (x−2)(x−1), the Galois group is again trivial.
• If it has two irrational roots (including the case where the roots are complex), then the Galois group contains two permutations, just as in the above example.

Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...

Second example — somewhat trickier

Consider the polynomial

x⁴ − 10 x² + 1,

which can also be written as

(x² − 5)² − 24.

We wish to describe the Galois group of this polynomial, again over the field of rational numbers. The polynomial has four roots: In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

A = √2 + √3,

B = √2 − √3,

C = −√2 + √3,

D = −√2 − √3.

There are 24 possible ways to permute these four roots, but not all of these permutations are members of the Galois group. The members of the Galois group must preserve any algebraic equation with rational coefficients involving A, B, C and D. One such equation is

A + D = 0.

Therefore the permutation

(A, B, C, D) → (A, B, D, C)

is not permitted, because it transforms the valid equation A + D = 0 into the equation A + C = 0, which is invalid since A + C = 2√3 ≠ 0.

Another equation that the roots satisfy is

(A + B)² = 8.

This will exclude further permutations, such as

(A, B, C, D) → (A, C, B, D).

Continuing in this way, we find that the only permutations remaining are

(A, B, C, D) → (A, B, C, D)

(A, B, C, D) → (C, D, A, B)

(A, B, C, D) → (B, A, D, C)

(A, B, C, D) → (D, C, B, A),

and the Galois group is isomorphic to the Klein four-group. This article is about the mathematical group. ...

The modern approach by field theory

In the modern approach, one starts with a field extension L/K, and examines the group of field automorphisms of L/K. See the article on Galois groups for further explanation and examples. In abstract algebra, a subfield of a field L is a subset K of L which is closed under the addition and multiplication operations of L and itself forms a field with these operations. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

The connection between the two approaches is as follows. The coefficients of the polynomial in question should be chosen from the base field K. The top field L should be the field obtained by adjoining the roots of the polynomial in question to the base field. Any permutation of the roots which respects algebraic equations as described above gives rise to an automorphism of L/K, and vice versa.

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. In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

There are several advantages to the modern approach over the permutation group approach.

• It permits a far simpler statement of the fundamental theorem of Galois theory.
• The use of base fields other than Q is crucial in many areas of mathematics. For example, in algebraic number theory, one often does Galois theory using number fields, finite fields or local fields as the base field.
• It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses the absolute Galois group of Q, defined to be the Galois group of K/Q where K is an algebraic closure of Q.
• It allows for consideration of inseparable extensions. This issue does not arise in the classical framework, since it was always implicitly assumed that arithmetic took place in characteristic zero, but nonzero characteristic arises frequently in number theory and in algebraic geometry.
• It removes the rather artificial reliance on chasing roots of polynomials. That is, different polynomials may yield the same extension fields, and the modern approach recognizes the connection between these polynomials.

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... In mathematics, the absolute Galois group of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. When K is a perfect field, Ksep is the same as an algebraic closure of K, and therefore this definition may be used for... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

Solvable groups and solution by radicals

The notion of a solvable group in group theory allows us to determine whether a polynomial is solvable in the radicals, depending on whether its Galois group has the property of solvability. In essence, each field extension L/K corresponds to a factor group in a composition series of the Galois group. If a factor group in the composition series is cyclic of order n, then the corresponding field extension is a radical extension, and the elements of L can then be expressed using the nth root of some element of K. In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, a composition series of a group G is a chain of subgroups of G satisfying where stands for normal subgroup, such that each quotient group Hi+1/Hi is a simple group. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

If all the factor groups in its composition series are cyclic, the Galois group is called solvable, and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually Q).

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—the Abel-Ruffini theorem. This is due to the fact that for n > 4 the symmetric group S_n contains a simple, non-cyclic, normal subgroup. The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

The inverse Galois problem

See main article: inverse Galois problem In mathematics, the inverse Galois problem concerns whether or not we can find a rational field extension with a given Galois group. ...

It is easy to construct field extensions with any given finite group as Galois group. That is, all finite groups do occur as Galois groups.

For that, 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 {x_α}, one for each element α of G, and adjoin them to K to get the field F = K({x_α}). Contained within F is the field L of symmetric rational functions in the {x_α}. 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. In group theory, Cayleys theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. ...

It is an open problem (in general) how to construct field extensions of a fixed ground field with a given finite group as Galois group.

External links

• http://www.partow.net/projects/galois/index.html Galois Field Arithmetic Library

Some on-line tutorials on Galois theory appear at:

• http://www.math.niu.edu/~beachy/aaol/galois.html
• http://nrich.maths.org/mathsf/journalf/feb02/art2/index_l2h.html

Online textbooks in French, German, Italian and English can be found at:

• http://www.galois-group.net/

References

• Emil Artin (1998). Galois Theory. Dover Publications. ISBN 0-486-62342-4. (Reprinting of second revised edition of 1944, The University of Notre Dame Press).
• Jacobson, Nathan (1985). Basic Algebra I (2nd ed). W.H. Freeman and Company. ISBN 0-7167-1480-9. (Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)
• Ian Stewart (1989). Galois Theory. Chapman and Hall. ISBN 0-412-34550-1.