Inverse Galois problem
Encyclopedia
In Galois theory
, the inverse Galois problem concerns whether or not every finite group
appears as the Galois group
of some Galois extension
of the rational number
s Q. This problem, first posed in the 19th century, is unsolved.
More generally, 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 one variable over the complex numbers C, and more generally over function fields in one variable over any algebraically closed field of characteristic
zero. Shafarevich showed that every finite solvable group
is realizable over Q. It also known that every sporadic group
, except possibly the Mathieu group
M23, is realizable over Q.
Hilbert
had shown 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
KG is rational over Q, then G is realizable over Q. Here rational means that it is a purely transcendental extension of Q, generated by an algebraically independent set. This criterion can for example be used to show that all the symmetric group
s are realizable.
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, starting with an extension of the field Q(t) of rational function
s 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.
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 Q(μ) be the cyclotomic extension of Q generated by μ, where μ is a primitive pth root of unity
; the Galois group of 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
has Galois group Z/nZ over Q. By taking appropriate sums of conjugates of μ, following the construction of Gaussian period
s, one can find an element α of F that generates F over Q, and compute its minimal polynomial.
This method can be extended to cover all finite abelian group
s, 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.)
Using the identity 1 + μ + μ2 + ... + μ6 = 0, one finds that
Therefore α is a root of the polynomial
which consequently has Galois group Z/3Z over Q.
showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients.
The polynomial has discriminantn(n−1)/2[nnbn−1 + (−1)1−n(n − 1)n−1an].
We take the special case
Substituting a prime integer for s in f(x,s) gives a polynomial (called a specialization of f(x,s)) that by Eisenstein's criterion
is irreducible. Then f(x,s) must be irreducible over Q(s). Furthermore, f(x,s) can be written
and f(x,1/2) can be factored to:(1 + 2x + 2x2 + ... + 2xn−1)/2
whose second factor is irreducible by Eisenstein's criterion. We have now shown that the group Gal(f(x,s)/Q(s)) is doubly transitive.
We can then find that this Galois group has a transposition. Use the scaling to get
and with get
which can be arranged to
Then g(y,1) has 1 as a double zero and its other n − 2 zeros are simple, and a transposition in Gal(f(x,s)/Q(s)) is implied. Any finite doubly transitive permutation group containing a transposition is a full symmetric group.
Hilbert's irreducibility theorem
then implies that an infinite set of rational numbers give specializations of f(x,t) whose Galois groups are Sn over the rational field Q. In fact this set of rational numbers is dense in Q.
The discriminant of g(y,t) equalsn(n−1)/2nn(n − 1)n−1tn−1(1 − t)
and this is not in general a perfect square.
Under this substitution the discriminant of g(y,t) equals
which is a perfect square when n is odd.
In the even case let t be the reciprocal of
and 1 − t becomes
and the discriminant becomes
which is a perfect square when n is even.
Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups.
and A be the set of n-tuples (g1,...gn) of G such that gi is in Ci and the product g1...gn is trivial. Then A is called rigid if it is nonempty, G acts transitively on it by conjugation, and each element of A generates G.
showed that if a finite group G has a rigid set then it can often be realized as a Galois group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible characters of G on the conjugacy classes Ci.)
This can be used to show that many finite simple groups, including the monster simple group, are Galois groups of extensions of the rationals.
The prototype for rigidity is the symmetric group Sn, which is generated by an n-cycle and a transposition whose product is an (n-1)-cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group.
PSL(2,Z), which is based on changes of basis for Λ. Let j denote the elliptic modular function of Klein. Define the polynomial φn as the product of the differences (X-j(Λi)) over the conjugate sublattices. As a polynomial in X, φn has coefficients that are polynomials over Q in j(τ).
On the conjugate lattices, the modular group acts as PGL(2,Zn). It follows that φn has Galois group isomorphic to PGL(2,Zn) over Q(J(τ)).
Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing φn to polynomials with Galois group PGL(2,Zn) over Q. The groups PGL(2,Zn) include infinitely many non-solvable groups.
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, the inverse Galois problem concerns whether or not every finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
appears as the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
of some Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
of the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s Q. This problem, first posed in the 19th century, is unsolved.
More generally, let G be a given finite group, and let K be a field. Then the question is this: is there a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
field L/K such that the Galois group of the extension is isomorphic
Group isomorphism
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. If there exists an isomorphism between two groups, then the groups are called isomorphic...
to G? One says that G is realizable over K if such a field L exists.
Partial results
There is a great deal of detailed information in particular cases. It is known that every finite group is realizable over any function fieldFunction field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
in one variable over the complex numbers C, and more generally over function fields in one variable over any algebraically closed field of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
zero. Shafarevich showed that every finite solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
is realizable over Q. It also known that every sporadic group
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
, except possibly the Mathieu group
Mathieu group
In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
M23, is realizable over Q.
Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
had shown 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
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
KG is rational over Q, then G is realizable over Q. Here rational means that it is a purely transcendental extension of Q, generated by an algebraically independent set. This criterion can for example be used to show that all the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
s are realizable.
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
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
: in algebraic terms, starting with an extension of the field Q(t) of rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s in an indeterminate t. After that, one applies Hilbert's irreducibility theorem
Hilbert's irreducibility theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers...
to specialise t, in such a way as to preserve the Galois group.
A simple example: cyclic groups
It is possible, using classical results, to construct explicitly a polynomial whose Galois group over Q is the cyclic groupCyclic 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 g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
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
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number 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. In other words, there are infinitely many primes which are...
. Let Q(μ) be the cyclotomic extension of Q generated by μ, where μ is a primitive pth root of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
; the Galois group of 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
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 its most basic form, the theorem asserts that given a field extension E /F which is finite and Galois, there is a one-to-one correspondence between its...
implies that the corresponding fixed field
has Galois group Z/nZ over Q. By taking appropriate sums of conjugates of μ, following the construction of Gaussian period
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
s, one can find an element α of F that generates F over Q, and compute its minimal polynomial.
This method can be extended to cover all finite abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s, 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
Kronecker–Weber theorem
In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
, which lies significantly deeper.)
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.
Symmetric and alternating groups
HilbertDavid Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients.
The polynomial has discriminantn(n−1)/2[nnbn−1 + (−1)1−n(n − 1)n−1an].
We take the special case
- f(x,s) = xn − sx − s.
Substituting a prime integer for s in f(x,s) gives a polynomial (called a specialization of f(x,s)) that by Eisenstein's criterion
Eisenstein's criterion
In mathematics, Eisenstein's criterion gives an easily checked sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers...
is irreducible. Then f(x,s) must be irreducible over Q(s). Furthermore, f(x,s) can be written
- xn − x/2 − 1/2 − (s − 1/2)(x + 1)
and f(x,1/2) can be factored to:(1 + 2x + 2x2 + ... + 2xn−1)/2
whose second factor is irreducible by Eisenstein's criterion. We have now shown that the group Gal(f(x,s)/Q(s)) is doubly transitive.
We can then find that this Galois group has a transposition. Use the scaling to get
- yn − s((1 − n)/n)n−1y − s((1 − n)/n)n
and with get
- g(y,t) = yn − nty + (n − 1)t
which can be arranged to
- yn − y − (n − 1)(y − 1) + (t − 1)(−ny + n − 1).
Then g(y,1) has 1 as a double zero and its other n − 2 zeros are simple, and a transposition in Gal(f(x,s)/Q(s)) is implied. Any finite doubly transitive permutation group containing a transposition is a full symmetric group.
Hilbert's irreducibility theorem
Hilbert's irreducibility theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers...
then implies that an infinite set of rational numbers give specializations of f(x,t) whose Galois groups are Sn over the rational field Q. In fact this set of rational numbers is dense in Q.
The discriminant of g(y,t) equalsn(n−1)/2nn(n − 1)n−1tn−1(1 − t)
and this is not in general a perfect square.
Alternating groups
Solutions for alternating groups must be handled differently for odd and even degrees. Let- t = 1 − (−1)n(n−1)/2nu2
Under this substitution the discriminant of g(y,t) equals
- nn+1(n − 1)n−1tn−1u2
which is a perfect square when n is odd.
In the even case let t be the reciprocal of
- 1 + (−1)n(n−1)/2(n − 1)u2
and 1 − t becomes
- t(−1)n(n−1)/2(n − 1)u2
and the discriminant becomes
- nn(n − 1)ntnu2
which is a perfect square when n is even.
Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups.
Rigid groups
Suppose that C1,...,Cn are conjugacy classes of a finite group G,and A be the set of n-tuples (g1,...gn) of G such that gi is in Ci and the product g1...gn is trivial. Then A is called rigid if it is nonempty, G acts transitively on it by conjugation, and each element of A generates G.
showed that if a finite group G has a rigid set then it can often be realized as a Galois group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible characters of G on the conjugacy classes Ci.)
This can be used to show that many finite simple groups, including the monster simple group, are Galois groups of extensions of the rationals.
The prototype for rigidity is the symmetric group Sn, which is generated by an n-cycle and a transposition whose product is an (n-1)-cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group.
A construction with an elliptic modular function
Let n be any integer greater than 1. A lattice Λ in the complex plane with period ratio τ has a sublattice Λ' with period ratio nτ. The latter lattice is one of a finite set of sublattices permuted by the modular groupModular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
PSL(2,Z), which is based on changes of basis for Λ. Let j denote the elliptic modular function of Klein. Define the polynomial φn as the product of the differences (X-j(Λi)) over the conjugate sublattices. As a polynomial in X, φn has coefficients that are polynomials over Q in j(τ).
On the conjugate lattices, the modular group acts as PGL(2,Zn). It follows that φn has Galois group isomorphic to PGL(2,Zn) over Q(J(τ)).
Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing φn to polynomials with Galois group PGL(2,Zn) over Q. The groups PGL(2,Zn) include infinitely many non-solvable groups.