**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[u_{1}, u_{2}, ..., u_{n}] means
A[{u_{1}, u_{2}, ..., u_{n}}].
Clearly if S is the union of S_{1} and S_{2}
then A[S] is (A[S_{1}])[S_{2}].

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 sub*fields*
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 S_{1} and S_{2}
then F(S) is (F(S_{1}))(S_{2}).

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.