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Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is the field F q ( x ) {\displaystyle \mathbf {F} _{q}(x)} , since the Frobenius endomorphism sends x ↦ x p {\displaystyle x\mapsto x^{p}} and therefore is not surjective.
The splitting field of x q − x over F p is the unique finite field F q for q = p n. [2] Sometimes this field is denoted by GF(q). The splitting field of x 2 + 1 over F 7 is F 49; the polynomial has no roots in F 7, i.e., −1 is not a square there, because 7 is not congruent to 1 modulo 4. [3]
An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree d for a polynomial, consider the expansion of n in base m (allowing digits between −m and m) for a number of different m of order n 1/d, and pick f(x) as the polynomial with the smallest coefficients and g(x) as x − m.
The map x ↦ L(x) is a linear map over any field containing F q.; The set of roots of L is an F q-vector space and is closed under the q-Frobenius map.; Conversely, if U is any F q-linear subspace of some finite field containing F q, then the polynomial that vanishes exactly on U is a linearised polynomial.
In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967.
Algorithm: SFF (Square-Free Factorization) Input: A monic polynomial f in F q [x] where q = p m Output: Square-free factorization of f R ← 1 # Make w be the product (without multiplicity) of all factors of f that have # multiplicity not divisible by p c ← gcd(f, f′) w ← f/c # Step 1: Identify all factors in w i ← 1 while w ≠ 1 do y ...
where h is a univariate polynomial in x 0 of degree D and g 0, ..., g n are univariate polynomials in x 0 of degree less than D. Given a zero-dimensional polynomial system over the rational numbers, the RUR has the following properties. All but a finite number linear combinations of the variables are separating variables.