Basic notations concerning ring and field extensions

All rings and fields will be assumed commutative unless stated otherwise.

Suppose B is some ring, A is a subring (a subset containing 0, 1 and closed under the ring operations), and S is any subset of B. We write A[S] for the subring generated by A and S; equivalently, the set of all elements of B obtained by evaluating on elements of S any polynomial over A (that is, with coefficients in A) in some (finite) number of variables. We use the shorthand: if u is an element of B then A[u] means A[{u}]; likewise, A[u1, u2, ..., un] means A[{u1, u2, ..., un}]. Clearly if S is the union of S1 and S2 then A[S] is (A[S1])[S2].

If B is unspecified, then A[S] is the ring of polynomials in |S| indeterminates. Note that this is consistent with our earlier use of this notation, and thus may be regarded as a special case. [In the general case, A[S] may be regarded as the image of the ring homomorphism from this polynomial ring to B that sends each element of A to itself and each indeterminate to the corresponding element of S.]

Now let K a field, and F a subfield. We then say that K is a field extension (sometimes also ``extension field'', or simply ``extension'') of F, and write ``F/K''. Since K is a vector space over itself, it is a fortiori a vector space over F; we write [K:F] for the (possibly infinite) dimension of this vector space.

Since in particular F is a subring of the ring K, we know what F[S], F[u], etc. mean. We also have the notations F(S), F(u), etc. for the subfields generated by F and S in K, or by indeterminates (``transcendentals'') if K is is not specified. The field F(S) consists of all quotients a/b with a,b in F[S] and b nonzero. Again it is clear that if S is the union of S1 and S2 then F(S) is (F(S1))(S2).

If an extension E of F can be written as F(u), we say that E a simple extension of F, with u as primitive element (or field generator of E/F). If u is not a root of any polynomial over F, we say that u is transcendental over F; else u is algebraic over F.