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The prime subfield of a field F is the subfield of F generated by the multiplicative identity 1_F of F. It is isomorphic to either Q (if the field characteristic is 0), or ...

The algebraic integers in a number field.

An element of an extension field of a field F which is not algebraic over F. A transcendental number is a complex number which is transcendental over the field Q of rational ...

A polynomial with coefficients in a field is separable if its factors have distinct roots in some extension field.

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

Let (K,|·|) be a non-Archimedean field. Its valuation ring R is defined to be R={x in K:|x|<=1}. The valuation ring has maximal ideal M={x in K:|x|<1}, and the field R/M is ...

The field F^_ is called an algebraic closure of F if F^_ is algebraic over F and if every polynomial f(x) in F[x] splits completely over F^_, so that F^_ can be said to ...

A theorem which treats constructions of fields of field characteristic p.

A field K is said to be algebraically closed if every polynomial with coefficients in K has a root in K.

An extension A subset B of a group, ring, module, field, etc., such that A!=B.