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One of the basic propositions required for completely determining the Galois group [3] of a finite field extension is the following: Given a polynomial () [], let / be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,
We conclude that the Galois group of the polynomial x 2 − 4x + 1 consists of two permutations: the identity permutation which leaves A and B untouched, and the transposition permutation which exchanges A and B. As all groups with two elements are isomorphic, this Galois group is isomorphic to the multiplicative group {1, −1}.
The Galois group of a polynomial of degree is or a proper subgroup of it. If a polynomial is separable and irreducible, then the corresponding Galois group is a transitive subgroup. Transitive subgroups of form a directed graph: one group can be a subgroup of several groups. One resolvent can tell if the Galois group of a polynomial is a (not ...
The resolvent cubic of an irreducible quartic polynomial P(x) can be used to determine its Galois group G; that is, the Galois group of the splitting field of P(x). Let m be the degree over k of the splitting field of the resolvent cubic (it can be either R 4 (y) or R 5 (y); they have the same splitting field).
In other words, the Galois group is A 3 if and only if the discriminant is the square of an element of K. As most integers are not squares, when working over the field Q of the rational numbers, the Galois group of most irreducible cubic polynomials is the group S 3 with six elements. An example of a Galois group A 3 with three elements is ...
Its Galois group = (/) comprises the automorphisms of K which fix a. Such automorphisms must send √ 2 to √ 2 or – √ 2, and send √ 3 to √ 3 or – √ 3, since they permute the roots of any irreducible polynomial.
A cubic field K is called a cyclic cubic field if it contains all three roots of its generating polynomial f. Equivalently, K is a cyclic cubic field if it is a Galois extension of Q , in which case its Galois group over Q is cyclic of order three.
The sum of a root and its conjugate is twice its real part. These three sums are the three real roots of the cubic polynomial +, and the primitive seventh roots of unity are , where r runs over the roots of the above polynomial. As for every cubic polynomial, these roots may be expressed in terms of square and cube roots.