Field theory

quotient of two integers

commutative ring in which every nonzero element is inversible

Every polynomial has a real or complex root

the absence of infinitesimals in a mathematical system

a finite degree (and hence algebraic) field extension of the field of rational numbers

number system for a prime p which extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems

element of a nonstandard model of the reals, which can be infinite or infinitesimal

in a field or a ring, the smallest positive integer, if any, such that the sum of n ones equals 0; zero otherwise

field in which every polynomial is factorizable into monomials

minimal field in which every polynomial in a given set decomposes into linear factors

smallest field in which an integral domain can be embedded

theorem

linearly independence of elements of a field extension that are also not related via finitary arithmetic operations

element of an extension field which is a root of a non-zero polynomial with coefficients in the subfield

algebraic number field of degree two over the rational numbers

given an element α of a field extension E/F, the unique monic polynomial p∈F[x] of minimal degree such that p(α) = 0

study of objects of arithmetic interest over infinite towers of number fields

function in algebra which generalises the concept of multiplicity for commutative rings

mathematical concept

class of extensions of the real numbers

a field that is either of characteristic 0, or of positive characteristic p such that every element admits a p-th root

non-discrete locally compact topological field

mathematical term; type of polynomial transformation

Field theory theorem

another root of the same minimal polynomial

expression whose number of distinct roots is equal to its degree

several notions

mathematical theory describing field extensions involving the adjunction of nth roots

subfield of K(X) is isomorphic to K(X)

Concept in algebra

Mathematical function

ordered field where every nonnegative element is a square

field extension that contains an element algebraically independent from the base field

field in which every sum of two squares is a square

minimal field extension containing at least one root of an irreducible polynomial

the minimal polynomial of a primitive element of a finite field regarded as a field extension over a finite field of prime order

a valuation on a field taking values in the integers

algebraic structure that is complete relative to a metric

finitely generated extension field of positive transcendence degree

a number field K such that, for each embedding of K into the complex numbers, the image lies inside the real numbers

theorem in differential algebra

the field of ratios of polynomial functions on a variety

field in which every element of the field has a square root in the field

field that can be equipped with an ordering