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Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomials over K, and indeed K(X) is the field of fractions of the polynomial ring K[X]. This field of rational functions is an extension field of K. This extension is infinite.
Radical extensions occur naturally when solving polynomial equations in radicals.In fact a solution in radicals is the expression of the solution as an element of a radical series: a polynomial f over a field K is said to be solvable by radicals if there is a splitting field of f over K contained in a radical extension of K.
Then the formal power series ring [[]] is completely integrally closed. [10] This is significant since the analog is false for an integrally closed domain: let R be a valuation domain of height at least 2 (which is integrally closed). Then [[]] is not integrally closed. [11] Let L be a field extension of K.
But since K has Krull dimension zero and since an integral ring extension (e.g., a finite ring extension) preserves Krull dimensions, the polynomial ring must have dimension zero; i.e., =. The following characterization of a Jacobson ring contains Zariski's lemma as a special case.
A Lubin–Tate extension of a local field K is an abelian extension of K obtained by considering the p-division points of a Lubin–Tate group. If g is an Eisenstein polynomial , f ( t ) = t g ( t ) and F the Lubin–Tate formal group, let θ n denote a root of gf n -1 ( t )= g ( f ( f (⋯( f ( t ))⋯))).
This definition is equivalent to saying that K has a unique (necessarily cyclic) extension K n of degree n for each integer n ≥ 1, and that the union of these extensions is equal to K s. [3] Moreover, as part of the structure of the quasi-finite field, there is a generator F n for each Gal( K n / K ), and the generators must be coherent , in ...
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 +, −, •, / .
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. Bourbaki calls such an extension a quasi-Galois extension.