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In this case, L is an extension field of K and K is a subfield of L. [1] [2] [3] 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.
A field extension L/K is called a simple extension if there exists an element θ in L with L = K ( θ ) . {\displaystyle L=K(\theta ).} This means that every element of L can be expressed as a rational fraction in θ , with coefficients in K ; that is, it is produced from θ and elements of K by the field operations +, −, •, / .
An extension of A by B is called split if it is equivalent to the trivial extension 0 → B → A ⊕ B → A → 0. {\displaystyle 0\to B\to A\oplus B\to A\to 0.} There is a one-to-one correspondence between equivalence classes of extensions of A by B and elements of Ext 1
In 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.
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.
Given a commutative ring A, an A-extension or an extension of an A-algebra is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules". An extension is said to be trivial or to split if ϕ {\displaystyle \phi } splits; i.e., ϕ {\displaystyle \phi } admits a section that is a ring homomorphism [ 2 ...
An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E. [10] An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice. [11]
In 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 linear factors in L. [ 1 ] [ 2 ] This is one of the conditions for an algebraic extension to be a Galois extension .