Galois theory

In mathematics, Galois theory is a branch of abstract algebra.

The permutation group approach to Galois theory

If we are given a polynomial, it may happen that some of the roots of the polynomial are connected by various algebraic equations. For example, it may turn out that for two of the roots, say A and B, the equation A2 + 5B3 = 7 holds. The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers. (One might instead specify a certain field in which the coefficients should lie, but for the simple examples below, we will restrict ourselves to the field of rational numbers.)

Related Topics:
Permutation - Rational number - Field

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These permutations together form a permutation group, also called the Galois group of the polynomial (over the rational numbers). This can be made much clearer by way of example.

Related Topics:
Permutation group - Galois group

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First example — a quadratic equation

Consider the quadratic equation

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:x2 − 4x + 1 = 0.

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By using the quadratic formula, we find that the two roots are

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:A = 2 + √3,   and

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:B = 2 − √3.

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Examples of algebraic equations satisfied by A and B include

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:A + B = 4,   and

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:AB = 1.

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Obviously, in either of these equations, if we exchange A and B, we obtain another true statement. For example, the equation A + B = 4 becomes simply B + A = 4. Furthermore, it is true, but far less obvious, that this holds for every possible algebraic equation satisfied by A and B; to prove this requires the theory of symmetric polynomials.

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We conclude that the Galois group of the polynomial x2 − 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 a group, it is isomorphic to the cyclic group of order two, denoted Z/2Z.

Related Topics:
Identity - Transposition - Isomorphic - Cyclic group

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One might raise the objection that A and B are related by yet another algebraic equation,

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:A − B − 2√3 = 0,

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which does not remain true when A and B are exchanged. However, this equation does not concern us, because it does not have rational coefficients; in particular, √3 is not rational.

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A similar discussion applies to any quadratic polynomial ax2 + bx + c, where a, b and c are rational numbers.

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• If the polynomial has only one root, for example x² − 4x + 4 = (x−2)², then the Galois group is trivial; that is, it contains only the identity permutation.
• If it has two distinct rational roots, for example x² − 3x + 2 = (x−2)(x−1), the Galois group is again trivial.
• If it has two irrational roots (including the case where the roots are complex), then the Galois group contains two permutations, just as in the above example.

Second example — somewhat trickier

Consider the polynomial

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:x4 − 10 x2 + 1,

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which can also be written as

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:(x2 − 5)2 − 24.

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We wish to describe the Galois group of this polynomial, again over the field of rational numbers. The polynomial has four roots:

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:A = √2 + √3,

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:B = √2 − √3,

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:C = −√2 + √3,

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:D = −√2 − √3.

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There are 24 possible ways to permute these four roots, but not all of these permutations are members of the Galois group. The members of the Galois group must preserve any algebraic equation (with rational coefficients!) involving A, B, C and D. One such equation is

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:A + D = 0.

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Therefore the permutation

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:(A, B, C, D) → (A, B, D, C)

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is not permitted, because it transforms the valid equation A + D = 0 into the equation A + C = 0, which is invalid since A + C = 2√3 ≠ 0.

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Another equation that the roots satisfy is

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:(A + B)2 = 8.

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This will exclude further permutations, such as

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:(A, B, C, D) → (A, C, B, D).

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Continuing in this way, we find that the only permutations remaining are

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:(A, B, C, D) → (A, B, C, D)

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:(A, B, C, D) → (C, D, A, B)

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:(A, B, C, D) → (B, A, D, C)

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:(A, B, C, D) → (D, C, B, A),

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and the Galois group is isomorphic to the Klein four-group.

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