algebraic extension
71Duality (mathematics) — In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one to one fashion, often (but not always) by means of an involution operation: if the dual… …
72Perfect field — In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds: Every irreducible polynomial over k has distinct roots. Every polynomial over k is separable. Every finite extension of k is separable. (This… …
73Quasi-algebraically closed field — In mathematics, a field F is called quasi algebraically closed (or C1) if for every non constant homogeneous polynomial P over F has a non trivial zero provided the number of its variables is more than its degree. In other words, if P is a non… …
74Cartesian coordinate system — Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2, 3) in green, (−3, 1) in red, (−1.5, −2.5) in blue, and the origin (0, 0) in purple. A Cartesian coordinate system specifies each point… …
75Fundamental theorem of Galois theory — In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.In its most basic form, the theorem asserts that given a field extension E / F which is finite and Galois,… …
76Normal closure — The term normal closure is used in two senses in mathematics: In group theory, the normal closure of a subset of a group is the smallest normal subgroup that contains the subset; see conjugate closure. In field theory, the normal closure of an… …
77Arithmetic function — In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that expresses some arithmetical property of n. [1] An example of an arithmetic… …
78Formally real field — In mathematics, a formally real field in field theory is a field that shares certain algebraic properties with the real number field. A formally real field F may be characterized in any of the following equivalent ways:* −1 is not a sum of… …
79Pseudo algebraically closed field — In mathematics, a field K is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds:*Each absolutely irreducible variety V defined over K has a K rational point. *Each absolutely irreducible… …
80Krull ring — A Krull ring is a particular type of commutative ring studied in commutative algebra and related branches of mathematics and named after the German mathematician Wolfgang Krull.Formal definitionLet A be an integral domain and let P be the set of… …
