Degree of a field extensionIn 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 role in many parts of mathematics, including algebra and number theory—indeed in any area where fields appear prominently. Definition and notationSuppose that E/F is a field extension. Then E may be considered as a vector space over F (the field of scalars).
Irreducible polynomialIn mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
Field extensionIn mathematics, particularly in algebra, a field extension is a pair of fieldsK⊆L{\displaystyle K\subseteq L}, such that the operations of K are those of Lrestricted to K. In this case, L is an extension field of K, and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
Abstract algebraIn mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field.
Degree of a polynomialIn mathematics, 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 is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.
Field (mathematics)Algebraic structure with addition, multiplication, and divisionA field is an algebraic structure that is closed under the four usual arithmetic operations (except when it comes to division by zero).In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do.
PolynomialType of mathematical expressionIn mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x{\displaystyle x} is x2−4x+7{\displaystyle x^{2}-4x+7}.IsomorphismThe group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as A≅B{\displaystyle A\cong B}.
Complex numberNumber with a real and an imaginary partA complex number z = a + bi can be visually represented as a pair of numbers (a, b) forming a position vector (blue) or a point (red) on an Argand diagram, representing the complex plane.
Ring homomorphismIn mathematics, a ring homomorphism is a structure-preserving function between two rings.
Zero of a functionIn mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued functionf{\displaystyle f}, is a member x{\displaystyle x} of the domain of f{\displaystyle f} such that f(x){\displaystyle f(x)}vanishes at x{\displaystyle x}; that is, the function f{\displaystyle f} attains the value of 0 at x{\displaystyle x}, or equivalently, x{\displaystyle x} is a solution to the equation f(x)=0{\displaystyle f(x)=0}.*As an Amazon Associate I earn from qualifying purchases.