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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 algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective. If K is a function field over k of n variables, and L is a function field in m variables, and n > m, then there are no morphisms from K to L.
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.
We can also assume without loss of generality that it is a reduced polynomial, because P(x) can be expressed as the product of two quadratic polynomials if and only if P(x − a 3 /4) can and this polynomial is a reduced one. Then R 3 (y) = y 3 + 2a 2 y 2 + (a 2 2 − 4a 0)y − a 1 2. There are two cases: If a 1 ≠ 0 then R 3 (0) = −a 1 2 < 0.
The Gaussian binomial coefficient, written as () or [], is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian (,).
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 ...
An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n such that F(x) divides x n − 1 is n = p m − 1. A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1, meaning that any of them generates the multiplicative group ...
The resulting polynomial is not a linear function of the coordinates (its degree can be higher than 1), but it is a linear function of the fitted data values. The determinant, permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns).