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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 (,).
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.
A polynomial f of degree n greater than one, which is irreducible over F q, defines a field extension of degree n which is isomorphic to the field with q n elements: the elements of this extension are the polynomials of degree lower than n; addition, subtraction and multiplication by an element of F q are those of the polynomials; the product ...
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. The set of linearised polynomials over a given field is closed under addition and composition of ...
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.
Specifically, for α ≠ 0 ∈ F q with q = p e for some prime p and any integers n ≥ 0 and 0 ≤ k < p, the n th Dickson polynomial of the (k + 1) th kind over F q, denoted by D n,k (x,α), is defined by [11]
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.
An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial. [1] If F is any field, the trivial extension is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.