In mathematics, the inverse Galois problem concerns whether or not we can find a rational field extension with a given Galois group. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

To be a little more precise, let G be a given finite group, and let K be a field. Then the question is this: is there a Galois extension field L/K such that the Galois group of the extension is isomorphic to G? One says that G is realizable over K if such a field L exists. In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, a Galois extension is a field extension that has a Galois group. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

This problem was first posed by Emmy Noether for the case where K is the field Q of rational numbers. There is a great deal of detailed information in particular cases. The problem is solved for function fields in one variable over the complex numbers C, and more generally for function fields in one variable over any algebraically closed field of characteristic zero. Shafarevich solved the problem for finite solvable groups in the case of Q. Emmy Noether (March 23, 1882 – April 14, 1935) was one of the most talented mathematicians of the early 20th century, with penetrating insights that she used to develop elegant abstractions which she formalized beautifully. ... 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. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... 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). ... Igor Rostislavovich Shafarevich (born 3 June 1923) is a Russian mathematician, founder of the major school of algebraic number theory and algebraic geometry in the USSR. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharovs Human Rights Committee from 1970. ... 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. ...

Hilbert had showed that this question is related to a rationality question for G: if K is any extension of Q, on which G acts as an automorphism group and the invariant field K^G is rational over Q, then G is realizable over Q. Here rational means that it is a pure transcendental extension of Q, generated by an algebraically independent set. This criterion can for example be used to show that all the symmetric groups are realizable. David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... In mathematics, a rationality question asks whether a given field extension is rational in the sense of algebraic geometry; such field extensions are also described as purely transcendental. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ... In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. This means that for every finite sequence α1,...,αn of elements of S, no two the... 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. ...

Much detailed work has been carried out on the question, which is in no sense solved in general. Some of this is based on constructing G geometrically as a Galois covering of the projective line: in algebraic terms, start with an extension of the field Q(t) of rational functions in an indeterminate t. After that, one applies Hilbert's irreducibility theorem to specialise t, in such a way as to preserve the Galois group. In mathematics, the projective line is a fundamental example of an algebraic curve. ... In mathematics, a rational function is a ratio of polynomials. ... In mathematics, Hilberts irreducibility theorem is a result of David Hilbert, stating that an irreducible polynomial in two variables and having rational number coefficients will remain irreducible as a polynomial in one variable, when a rational number is substituted for the other variable, in infinitely many ways. ...

A simple example: cyclic groups

It is possible, using classical results, to construct explicitly a polynomial whose Galois group over Q is the cyclic 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. ...

Z/nZ

for any positive integer n. To do this, choose a prime p such that

p ≡ 1 (mod n);

this is possible by Dirichlet's theorem. Let In number theory, Dirichlets theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n > 0, or in other words: there are infinitely many primes which are congruent to a modulo d. ...

Q(μ)

be the cyclotomic extension of Q generated by μ, where μ is a primitive p^th root of unity; the Galois group of In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...

Q(μ)/Q

is cyclic of order

p − 1.

Since n divides p − 1, the Galois group has a cyclic subgroup H of order

(p − 1)/n.

The fundamental theorem of Galois theory implies that the corresponding fixed field In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. ...

F = Q(μ)^H

has Galois group

Z/nZ

over Q. By taking appropriate sums of conjugates of μ, following the construction of Gaussian periods, one can find an element α of F that generates F over Q, and to compute its minimal polynomial. In mathematics, a Gaussian period is a certain kind of sum of roots of unity. ...

This method can be extended to cover all finite abelian groups, since every such group appears in fact as a quotient of the Galois group of some cyclotomic extension of Q. (This statement should not though be confused with the Kronecker-Weber theorem, which lies significantly deeper.) In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In algebraic number theory, the Kronecker-Weber theorem states that every finite abelian extension of the field of rational numbers , or in other words every algebraic number field whose Galois group over is abelian, is a subfield of a cyclotomic field, i. ...

Worked example: the cyclic group of order three

For n = 3, we may take p = 7. Then Gal(Q(μ)/Q) is cyclic of order six. Let us take the generator η of this group which sends μ to μ³. We are interested in the subgroup H = {1, η³} of order two. Consider the element α = μ + η³(μ). By construction, α is fixed by H, and only has three conjugates over Q, given by

α = μ + μ⁶,    β = η(α) = μ³ + μ⁴,    γ = η²(α) = μ² + μ⁵.

Using the identity 1 + μ + μ² + ... + μ⁶ = 0, one finds that

α + β + γ = −1,

αβ + βγ + γα = −2, and

αβγ = 1.

Therefore α is a root of the polynomial

(x − α)(x − β)(x − γ) = x³ + x² − 2x − 1,

which consequently has Galois group Z/3Z over Q.

References

• Gunter Malle, Heinrich Matzat, Inverse Galois Theory, ISBN 3-540-62890-8.
• Alexander Schmidt, Kay Wingberg, Safarevic's Theorem on Solvable Groups as Galois Groups (see also Jürgen Neukirch, Alexander Schmidt, Kay Wingberg, Cohomology of Number Fields, Springer-Verlag, 1999, ISBN 3-540-66671-0.)