Ground field, splitting field, root field, Galois group, solvable group

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Ground field of an equation. The ground field of the polynomial equation of degree n

1) xⁿ + a₁x^n-1 + .... + a_n = 0

whose coefficients are assumed to have given values (as, for example, certain complex numbers) is the field consisting of the set of all quantities that can be obtained from the coefficients of the equation by means of a finite number of the operations of addition, subtraction, multiplication, and division.

Syn. Domain of rationality

Example. If the equation has rational coefficients, the ground field consists of all rational numbers. If the equation has the form

then the ground field consists of all numbers of the form

where a and b are rational numbers.

Splitting field (or root field) of a polynomial. For a polynomial p with coefficients in a field F, the splitting field F* is the smallest field extension of F over which the polynomial splits or decomposes into linear factors.

Syn root field, Galois field

Suppose the polynomial equation

xⁿ + a₁x^n-1 + .... + a_n = 0

has the roots ξ₁, ξ₂, .... , ξ_n. Then the splitting field (or root field) of the polynomial consists of the set of all quantities that can be obtained from ξ₁, ξ₂, .... , ξ_n by means of a finite number of the operations of addition, subtraction, multiplication, and division.

Examples.

1. The splitting field (or root field) of the equation x² + 1 = 0 is the set of complex numbers a + bi with rational a,b.

2. The splitting field (or root field) of the equation

is the set of numbers of the form

Through Viete’s formulas the coefficients of a polynomial equation are obtained from its roots by means of the operation of addition and multiplication so the root field of an equation must always contain its ground field. See Fig. 1. Sometimes the two fields coincide.

Automorphism of the splitting field with respect to the ground field. Let P be the ground field and K be the splitting field of a polynomial. A one-to-one mapping A of the splitting field K onto itself is called an automorphism of the splitting field K with respect to the ground field P if for every pair of elements of the splitting field their sum goes over into the sum of their images, and their product into the product, and every element of the ground field goes over into itself. These properties can be described by the formulas

(a + b)A = aA + bA, (ab)A = aA⋅bA αA = α (a,b ∈ K, α ∈ P)

where aA is the image of a: that is, aA is the element into which a goes over under the mapping A.

Theorem 1. The set of all automorphisms of the root field relative to the ground field is a group.

Galois group. The Galois group consists of the set of all automorphisms of the root field relative to the ground field.

Theorem 2. An automorphism of the Galois group carries a root of the given polynomial equation into another root. Proof

Thus every automorphism A effects a definite permutation of the roots of the equation. The set of all automorphisms of a polynomial consists of all possible mappings of the roots of the polynomial onto each other.

The Galois group is isomorphic to the group of permutations of the zeros of p.

Galois theory. A theory of the Galois field F* and the Galois group of a polynomial p with coefficients in a field F. The theory involves a one-to-one correspondence between the subfields of F* that contain F and the subgroups of the Galois group (the field K corresponds to the group G if and only if K is the set of members x of F* for which a(x) = x if a is in G, or if and only if G is the set of all automorphisms a of F* for which a(x) = x if x is in K). This leads to the theorem that the Galois group of a polynomial p with respect to a field F is solvable if the equation p(x) = 0 is solvable in F by radicals, from which it follows that there is a real quintic equation that is not solvable by radicals.

Solvable group. A group that contains a sequence of invariant subgroups, beginning with itself and ending with the identity, such that: (1) Each invariant subgroup is contained in the preceding one; (2) the quotient of the order of any one of the invariant groups by the order of the following one is a prime integer.

Finding the Galois group of an equation. Finding the Galois group of a given equation is usually a complicated problem. It is comparatively easy only in special cases.

Consider the equation

2) xⁿ + a₁x^n-1 + .... + a_n = 0

with literal coefficients a₁, a₂, .... a_n. The ground field of this equation is formed by the rational fractions of the coefficients, i.e. the fractions whose numerators and denominators are polynomials in a₁, a₂, .... a_n. The splitting field is formed by the rational fractions of the roots ξ₁, ξ₂, .... , ξ_n of the equation. These roots ξ₁, ξ₂, .... , ξ_n are connected with the coefficients by the formulas

-a₁ = ξ₁+ ξ₂ + .... + ξ_n

3) a₂ = ξ₁ξ₂+ ξ₁ξ₃ + .... + ξ_n-1ξ_n

................................................

(-1)ⁿa_n = ξ₁ξ₂.... ξ_{n .}

Because equation 2) is general, we can regard its roots as independent variables. Every permutation of these roots gives rise to an automorphism of the splitting field. Due to the symmetry of formulas 3) it can be seen that under every permutation of the roots the coefficients go over into themselves and, together with them, all rational fractions formed from them also go over into themselves. Thus, the Galois group of the general equation of degree n is essentially the symmetric group of all permutations of n letters.

References

Mathematics, Its Content, Methods and Meaning. Chap. XX