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prime subfield (Definition)

The prime subfield of a field $ F$ is the intersection of all subfields of $ F$, or equivalently the smallest subfield of $ F$. It can also be constructed by taking the quotient field of the additive subgroup of $ F$ generated by the multiplicative identity $ 1$.

If $ F$ has characteristic $ p$ where $ p > 0$ is a prime, then the prime subfield of $ F$ is isomorphic to the field $ \mathbb{Z}/p\mathbb{Z}$ of integers mod $ p$. When $ F$ has characteristic zero, the prime subfield of $ F$ is isomorphic to the field $ \mathbb{Q}$ of rational numbers.



"prime subfield" is owned by djao.
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proof that Q is the prime subfield of any field of characteristic 0 (Proof) by CWoo
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Cross-references: rational numbers, integers, isomorphic, prime, characteristic, multiplicative identity, generated by, subgroup, additive, quotient field, subfields, intersection, field
There are 11 references to this entry.

This is version 1 of prime subfield, born on 2002-05-03.
Object id is 2892, canonical name is PrimeSubfield.
Accessed 4134 times total.

Classification:
AMS MSC: 12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)
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