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In mathematics, a Galois extension is an algebraic
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
Aut(''E''/''F'') is precisely the base
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''F''. The significance of being a Galois extension is that the extension has a
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
and obeys the
fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basic ...
. A result of
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...
allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'' with fixed field ''F'', then ''E''/''F'' is a Galois extension.


Characterization of Galois extensions

An important theorem of
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...
states that for a
finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — ...
E/F, each of the following statements is equivalent to the statement that E/F is Galois: *E/F is a
normal extension In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic ext ...
and a
separable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polyn ...
. *E is a
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polyn ...
of a
separable polynomial In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely ...
with coefficients in F. *, \!\operatorname(E/F), = :F that is, the number of automorphisms equals the degree of the extension. Other equivalent statements are: *Every irreducible polynomial in F /math> with at least one root in E splits over E and is separable. *, \!\operatorname(E/F), \geq :F that is, the number of automorphisms is at least the degree of the extension. *F is the fixed field of a subgroup of \operatorname(E). *F is the fixed field of \operatorname(E/F). *There is a one-to-one correspondence between subfields of E/F and subgroups of \operatorname(E/F).


Examples

There are two basic ways to construct examples of Galois extensions. * Take any field E, any finite subgroup of \operatorname(E), and let F be the fixed field. * Take any field F, any separable polynomial in F /math>, and let E be its
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polyn ...
. Adjoining to the
rational number field In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...
gives a Galois extension, while adjoining the cubic root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first of them is the splitting field of x^2 -2; the second has normal closure that includes the complex cubic roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and x^3 -2 has just one real root. For more detailed examples, see the page on the
fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basic ...
. An algebraic closure \bar K of an arbitrary field K is Galois over K if and only if K is a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' i ...
.


Notes


Citations


References

*


Further reading

* * * ''(Galois' original paper, with extensive background and commentary.)'' * * * ''(Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)'' * (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois
groupoids In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial funct ...
.) * * * * *. English translation (of 2nd revised edition): ''(Later republished in English by Springer under the title "Algebra".)'' * {{DEFAULTSORT:Galois Extension Galois theory Algebraic number theory Field extensions