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In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules.
The inverse Galois problem is to find a field extension with a given Galois group. As long as one does not also specify the ground field, the problem is not very difficult, and all finite groups do occur as Galois groups. For showing this, one may proceed as follows. Choose a field K and a finite group G.
Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). [1] More narrowly, a Galois geometry may be defined as a projective space over a finite field. [2] Objects of study include affine and ...
The field E H is a normal extension of F (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H is a normal subgroup of Gal(E/F). In this case, the restriction of the elements of Gal(E/F) to E H induces an isomorphism between Gal(E H /F) and the quotient group Gal(E/F)/H.
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory , so named in honor of ...
The finite field with p n elements is denoted GF(p n) and is also called the Galois field of order p n, in honor of the founder of finite field theory, Évariste Galois.GF(p), where p is a prime number, is simply the ring of integers modulo p.
This field is called a finite field or Galois field with four elements, and is denoted F 4 or GF(4). [8] The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F 2 or GF(2) .
The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. [a] A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension. [2 ...