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Finite Extension

A finite extension is an extension field such that the dimension of as a vector space over (the so-called degree of over ) is finite. A finite field extension is always algebraic.

Note that "finite" is a synonym for "finite-dimensional"; it does not mean "of finite cardinality" (the field of complex numbers is a finite extension, of degree 2, of the field of real numbers, but is obviously an infinite set), and it is not even equivalent to "finitely generated" (a transcendental extension is never a finite extension, but it can be generated as a field by a single element as, for example, the field of rational functions over a field ).

A ring extension is called finite if is finitely generated as a module over . An example is the ring of Gaussian integers , which is generated by as a module over . The polynomial ring , however, is not a finite ring extension of , because the polynomial degrees of any finite set of polynomials are bounded. Their -linear combinations have the same bound and cannot include all powers of . The simplest generating set is the sequence .

A finite ring extension of commutative rings with the same multiplicative identity is an integral extension. More generally, such an extension ring is finite iff it is integral and is a finitely generated algebra over (Stacks Project, Tag 00GH). Thus finite generation as an algebra is weaker than finite generation as a module. A finite algebra allows a structure ring homomorphism that need not be injective, so it need not be an extension ring in the embedded sense.