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In the most general sense, ''Galois theory''
is a theory dealing with mathematical
objects on the basis of their automorphism groups. For instance,
Galois theories of fields, rings, topological spaces, etc., are
possible. In a narrower sense Galois theory is the Galois theory of
fields. The theory originated in the context of finding roots of
algebraic equations of high degrees. The familiar formula for solving
equations of degree two dates back to early Antiquity. Methods for
solving cubic (cf.
[[Cardano formula|Cardano formula]]) and quartic (cf.
[[Ferrari method|Ferrari method]]) equations were discovered in the
16th century. Unsuccessful attempts to find formulas for solving
quintic and higher-degree equations were made during the three
centuries which followed. It was finally proved by N.H. Abel in 1824
that there are no solutions in radicals of the general equation of
degree $\ge 5$. The next problem was to find necessary and sufficient
conditions to be satisfied by the coefficients of an equation for the
latter to be solvable in radicals, i.e. for it to be reducible to a
chain of two-term equations of the form $x^n-a=0$. This problem was solved
by E. Galois; his results were exposed in a letter on the eve of his
death (1832), and published in 1846. The theory of Galois will now be
summarized in modern language.

Let $k$ be an arbitrary field. An extension of $k$ is any field $K$
that contains $k$ as a subfield. Any extension may be regarded as a
linear space over $k$; if this space has finite dimension $n$, the
extension is called finite, while the dimension $n$ is called the
degree of the extension. An element $\def\a{\alpha}\a$ of an extension of $k$ is said
to be algebraic over $k$ if it is the root of an equation $f=0$, where
$f$ is a non-zero polynomial with coefficients from $k$ (this
polynomial may be taken to be irreducible). The smallest extension of
$k$ containing the algebraic element $\a$ over $k$ is usually denoted
by $k(\a)$. A finite extension $K$ of $k$ is called separable if $K=k(\a)$ and
if the irreducible polynomial $f$ with $\a$ as a root has no multiple
roots. If the characteristic of $k$ is zero (e.g. if $k$ is a number
field), any finite extension is separable (the theorem on the
primitive element). The splitting field of an irreducible polynomial
$f$ in $k[X]$ is the smallest extension of $k$ that contains all the
roots of this polynomial. The degree of such an extension is divisible
by the degree of $f$ and is equal to it if all the roots of $f$ can be
expressed as polynomials in one of them. An extension $K$ of $k$ is
called normal if it is the splitting field of a certain polynomial in
$k[X]$, and is called a Galois extension if it is normal and
separable. The group of all automorphisms of a Galois extension $K$
that leave all elements of $k$ invariant is called the Galois group of
this extension and is denoted by $\def\Gal{\textrm{Gal}}\Gal(K/k)$. Its order (the number of
elements) is equal to the degree of $K$ over $k$. To each subgroup $H$
of $\Gal(K/k)$ corresponds a subfield $P$ of $K$, consisting of all elements
from $K$ that remain fixed under all automorphisms from
$H$. Conversely, to each subfield $P\subset K$ that contains $k$ corresponds a
subgroup $H$ of $\Gal(K/k)$. It consists of all automorphisms leaving each
element of $P$ invariant. Here, $K$ is a Galois extension of $P$ and
$\Gal(K/P)=H$. The main theorem in Galois theory states that these
correspondences are mutually inverse, and are therefore one-to-one
correspondences between all subgroups of $\Gal(K/k)$ and all subfields of $K$
containing $k$. Thus, the description of all subfields of $K$ is
reduced to the description of all subgroups of the finite group $\Gal(K/k)$,
which is a much simpler task. It is important to note that in this
correspondence certain "good" properties of subgroups correspond to
the "good" properties of subfields and vice versa. Thus, a subgroup
$H$ will be a normal subgroup of $\Gal(K/k)=G$ if and only if the field $P$
which corresponds to it is a Galois extension of $k$. Moreover, $\Gal(P/k)$ is
isomorphic to $G/H$. To each ascending chain 
$$k=K_0\subset K_1\subset\cdots\subset K_r=K\label{1}$$
of subfields of $K$
corresponds a descending chain 
$$G=H_0\supset H_1\supset\cdots\supset H_r=\{e\}\label{2}$$
of subgroups of $G$, where
$H_i=\Gal(K/K_i)$. The chain (2) is a normal series (i.e. each group $H_{i+1}$ is a normal
subgroup of $H_i$ if $0\le i < r$) if and only if each field $K_{i+1}$ in the chain (1)
is a Galois extension of $K_i$, and in this a case one has $H_i/H_{i+1} \cong \Gal(K_{i+1}/K_i)$.

These results are applied to the solution of algebraic equations as
follows. Let $f$ be an irreducible polynomial without multiple roots
over the field $k$ and let $K$ be its splitting field (which will be a
Galois extension of $k$). The Galois group of this extension is called
the Galois group of the equation $f=0$. Solving the equation $f=0$ is
reduced to solving a chain of equations $f_1=0,\dots,f_r=0$ if and only if $K$ is
contained in a field $\bar K$ that is the last term of the ascending chain
of fields 
$$k=K_0\subseteq K_1\subseteq\cdots\subseteq K_r=K,$$
where $K_i$, $i=1,\dots,r$, is the splitting field of $f_i$ over
$K_{i-1}$. This last condition is equivalent with the group $G=\Gal(K/k)$ being a
quotient group of the group $\bar G=\Gal(\bar K/k)$ with a normal series whose factors $H_i/H_{i+1}$
are isomorphic to the Galois groups of the equations $f_i=0$ over $K_{i-1}$.

Let the field $k$ contain all roots of unity of order $n$. Then, for
any $a\in k$, the splitting field of the polynomial $x^n-a$ is $k(\a)$, where $\a$
is one of the values of the radical $a^{1/n}$. In such a case $\Gal(k(\a)/k)$ is a
cyclic group of order dividing $n$; conversely, if $\Gal(K/k)$ is a cyclic
group of order $n$, one has $K=k(\a)$, where $\a$ is the root of some
two-term equation $x^n-a = 0$. Thus, if $k$ contains the roots of unity of all
possible orders, then the equation $f=0$ is solvable in radicals if and
only if its Galois group is solvable (i.e. has a normal series with
cyclic factors $H_i/H_{i+1}$). This condition of solvability in radicals is also
valid if $k$ does not contain all possible roots of unity, since the
Galois group $\Gal(k'/k)$ of the extension $k'$ which is obtained by adjoining
these roots is always solvable.

In practical applications of the solvability condition, a very
important fact is that the Galois group of an equation can be computed
without solving the equation itself. The idea of this computation can
be stated as follows. Each automorphism of the splitting field of a
polynomial $f$ induces a permutation of its roots, and the
automorphism is completely determined by this permutation. For this
reason the Galois group of the equation may be treated, in principle,
as a subgroup of the group of permutations of its roots (namely, the
subgroup consisting of permutations that preserve all algebraic
relations between the roots). The relations between the roots of a
polynomial yield certain relations between its coefficients (by virtue
of Viète's formulas); by analyzing these relations it is possible to
determine the relations between the roots of the polynomial and thus
to compute the Galois group of the equation. In the general case the
Galois group of an algebraic equation consists of all permutations of
the roots, i.e. is the symmetric group of degree $n$. Since a
symmetric group is unsolvable for $n\ge 5$, there are, generally speaking,
no solutions in radicals of quintic equations and equations of a
higher degree (Abel's theorem).

The ideas of Galois theory permit, in particular, to give a complete
description of the class of construction problems that are solvable by
ruler and compass. It is possible to show by methods of analytic
geometry that any such construction problem can be reduced to some
algebraic equation over the field of rational numbers, and the problem
is solvable by using a ruler and compass if and only if the
corresponding equation is solvable in quadratic radicals. This is the
case if and only if the Galois group of the equation has a normal
series whose factors are groups of order two, and this occurs if and
only if its order is a power of two. Thus, a construction problem
solvable by ruler and compass is reduced to solving an equation whose
splitting field has degree $2^s$ over the field of rational numbers for
some $s$; if the degree of the equation is not of the form $2^s$, such a
construction is impossible. This is the case for the problem of
doubling the cube (which leads to the cubic equation $x^3-2 = 0$) and for the
problem of trisecting a given angle (which also leads to a cubic
equation). If $p$ is a prime number, the problem of constructing a
regular $p$-gon leads to the equation $x^{p-1}+\cdots + x + 1 = 0$, whose splitting field is
generated by any root and is therefore of degree $p-1$, the degree of
the equation. In this case a ruler-and-compass construction is
possible if and only if $p=2^s+1$ (e.g. if $p=5$ and $p=17$ it is possible, but
is not possible for $p=7$ or $p=13$).

The ideas of Galois had a decisive influence on the development of
algebra during almost a whole century. Galois theory was extended and
generalized in many directions. W. Krull developed a Galois theory for
infinite extensions (cf.
[[Galois topological group|Galois topological group]]); it was proved
(the [[Kronecker–Weber theorem]]) that the roots of an equation with
rational coefficients and with an Abelian Galois group are rational
linear combinations of roots of unity; a classification of the Abelian
extensions of a given algebraic number field (class field theory) was
given; a proof was given for the existence of an algebraic number
field with given solvable Galois group over the field of rational
numbers (cf.
[[Galois theory, inverse problem of|Galois theory, inverse problem
of]]). Nevertheless, classical Galois theory still contains many
unsolved problems. For instance, it is not known whether any finite
group occurs as the Galois group of an equation over the field of
rational numbers.

====References====
{|
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|valign="top"|{{Ref|Ar}}||valign="top"| E. Artin, "Galois theory", Notre Dame Univ., Indiana (1942)     {{MR|0006974}}  {{ZBL|0060.04813}} 
|-
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms", '''2''', Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)      
|-
|valign="top"|{{Ref|Ch}}||valign="top"| N.G. Chebotarev, "Galois theory", Moscow-Leningrad (1936) (In Russian)      
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|valign="top"|{{Ref|Ga}}||valign="top"| E. Galois, "Écrits et mémoires d'E. Galois", Gauthier-Villars (1962)      
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|valign="top"|{{Ref|Ko}}||valign="top"| H. Koch, "Galoissche Theorie der $p$-Erweiterungen", Deutsch. Verlag Wissenschaft.  (1970)  {{MR|0291139}}  {{ZBL|0216.04704}}    
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|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Algebra", Addison-Wesley (1974)   {{ZBL|0283.15001}}    
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|valign="top"|{{Ref|Po}}||valign="top"| M.M. Postnikov, "Fundamentals of Galois theory", Noordhoff (1962) (Translated from Russian)  {{MR|0136603}}     
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|valign="top"|{{Ref|Po2}}||valign="top"| M.M. Postnikov, "Fundamentals of Galois theory", Noordhoff (1962) (Translated from Russian)  {{MR|0136603}}     
|-
|valign="top"|{{Ref|Ts}}||valign="top"| N.G. [N.G. Chebotarev] Tschebotaröw, "Grundzüge der Galois'schen Theorie", Noordhoff (1950) (Translated from Russian)      
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