nLab free field

This entry is about the concept in algebra. For the concept of the same name in physics see at free field theory.

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Higher algebras

• symmetric monoidal (∞,1)-category of spectra

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Idea

Any commutative unital domain can be embedded into a smallest field in a unique way, its field of fractions. For noncommutative domains, both the existence and uniqueness (of a smallest skew field) do not need to hold, and are difficult to decide.

A classical theorem guarantees the existence for a class of Ore domains that is the domains for which the subset of regular elements is an Ore subset?; then the corresponding Ore localization is a skewfield and the localization map is an injection. Another important case is the case of free associative algebras; in that case the solution is not unique and the first embedding was constructed by Amitsur, and by a different procedure by P. M. Cohn. This skewfield is called the free field. It has been more recently used in formulating the theory of quasideterminants.

I've moved this to simply free field, on the grounds that (as with field itself) the algebraic mathematical meaning is likely to be our default here. But I might be wrong. —Toby

Last revised on November 19, 2015 at 07:32:11. See the history of this page for a list of all contributions to it.