mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important...

particularly in algebra, a field extension is a pair of fields K ⊆ L {\displaystyle K\subseteq L} , such that the operations of K are those of L restricted to...

exponent Degree of a field extension Degree of an algebraic number field, its degree as a field extension of the rational numbers Degree of an algebraic...

transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental...

In field theory, a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle...

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 field extension K / Q {\displaystyle K/\mathbb {Q} } has finite degree (and hence is an algebraic field extension). Thus K {\displaystyle K} is a field...

one of two rules in mathematics: Law of total expectation, in probability and stochastic theory a rule governing the degree of a field extension of a field...

In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a primitive element. Simple extensions...

extension (and hence an algebraic extension) of K. The field L is then a finite-dimensional vector space over K. Multiplication by α, an element of L...

Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers...

proper algebraic extension K, then the minimal polynomial of an element in K \ F is irreducible and its degree is greater than 1. The field F is algebraically...

(the degree of the extension of residue fields). It is also less than or equal to the degree of L/K. When L/K is separable, the ramification index of w over...

coordinates of previous points, of no greater degree than a quadratic. This implies that the degree of the field extension generated by a constructible...

mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by...

then E is an extension field of F. We then also say that E/F is a field extension. Degree of an extension Given an extension E/F, the field E can be considered...

algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every element of L...

field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree n...

include the field of study rather than the ambiguous 'Extension Studies.'" While the school retains "Extension Studies" in official degree titles, transcripts...

In field theory, a branch of mathematics, the minimal polynomial of an element α of an extension field of a field is, roughly speaking, the polynomial...

F being a field extension (or just extension) of E, denoted by F / E, and read "F over E". A basic datum of a field extension is its degree [F : E],...

field extension: if F is an extension of a field E, then [ F : E ] {\displaystyle [F:E]} denotes the degree of the field extension F / E {\displaystyle F/E}...

abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into...

for extensions of prime degree p, and Witt (1936) generalized it to extensions of prime power degree pn. If K is a field of characteristic p, a prime...

mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was...

a linear subspace of the finite-degree field extension Q ( α , β ) {\displaystyle \mathbb {Q} (\alpha ,\beta )} , and therefore has a finite degree itself...

the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term...

algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation...

bachelor's degree program in the same field, or as part of an integrated honours program. Programs like these typically require completion of a full year-long...

a field extension KX(U) of k. The dimension of U will be equal to the transcendence degree of this field extension. All finite transcendence degree field...