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Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, (x – a)(x – b) = x 2 – (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.
Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
Abel–Ruffini theorem. In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.
One of the basic propositions required for completely determining the Galois groups [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,
If all these trinomials are reducible, one chooses "pentanomials" X n + X a + X b + X c + 1, as polynomials of degree greater than 1, with an even number of terms, are never irreducible in characteristic 2, having 1 as a root. [3] A possible choice for such a polynomial is given by Conway polynomials. They ensure a certain compatibility between ...
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 relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is ...
More generally, given a commutative ring R (not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and Zariski closed subsets of the affine variety Spec(R). More generally, there is an antitone Galois connection between ideals in the ring and subschemes of the corresponding affine variety.