Galois theory

2020 Mathematics Subject Classification: Primary: 12Fxx [MSN][ZBL]

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) and quartic (cf. 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); 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). 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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[Ko]	H. Koch, "Galoissche Theorie der $p$-Erweiterungen", Deutsch. Verlag Wissenschaft. (1970) MR0291139 Zbl 0216.04704
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