Hilbert Field (other)
Hilbert field may refer to: * The Hilbert field, the minimal ordered Pythagorean field * A Hilbert field is one with minimal Kaplansky radical * Hilbert class field, the maximal abelian unramified extension of a number field * Hilbert–Speiser field, a field with a normal integral basis * Hilbertian field In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundam ...

Pythagorean Field
In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lambda in F. So a Pythagorean field is one closed under taking Pythagorean extensions. For any field F there is a minimal Pythagorean field F^ containing it, unique up to isomorphism, called its Pythagorean closure.Milnor & Husemoller (1973) p. 71 The ''Hilbert field'' is the minimal ordered Pythagorean field.Greenberg (2010) Properties Every Euclidean field (an ordered field in which all non-negative elements are squares) is an ordered Pythagorean field, but the converse does not hold.Martin (1998) p. 89 A quadratically closed field is Pythagorean field but not conversely (\mathbf is Pythagorean); however, a non formally real Pythagorean field is quadratically closed.Rajwade (1993) p.230 The Witt ring of a Pythagorean field is ...
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Kaplansky Radical
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields. Quadratic Hilbert symbol Over a local field ''K'' whose multiplicative group of non-zero elements is ''K''×, the quadratic Hilbert symbol is the function (–, –) from ''K''× × ''K''× to defined by :(a,b)=\begin+1,&\mboxz^2=ax^2+by^2\mbox(x,y,z)\in K^3;\\-1,&\mbox\end Equivalently, (a, b) = 1 if and only if b is equal to the norm of an element of the quadratic extension Ksqrt/math> page 110. Properties The follow ...
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Hilbert Class Field
In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonically isomorphic to the ideal class group of ''K'' using Frobenius elements for prime ideals in ''K''. In this context, the Hilbert class field of ''K'' is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every real embedding of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E''). Examples *If the ring of integers of ''K'' is a unique factorization domain, in particular if K = \mathbb , then ''K'' is its own Hilbert class field. *Let K = \mathbb(\sqrt) of discriminant -15. The field L = \mathbb(\sqrt, \sqrt) has discriminant 225=(-15)^2 and so is an everywhere unramified extension of ''K'', and it is abelian. ...
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