Subfields and Field Extensions

Before start to know about Subfields  we have to think some thing more about Field .

In case   F  be a field then the set of all Polynomials   F[x]   is an an integral domain .

 Choose   f(x) ,  g(x)  to any two elements of   F[x]  ,  then   f(x)  is said to be a divisor of   g(x)  if there exists a polynomial   h(x)   in   F[x]  such that .

                                                     g(x)  =  f(x) h(x)  .

The statement   f(x)   is a divisor of   g(x)  is expressed symbolically as   f(x)/g(x)   .

In case   f(x)/g(x)  and   g(x)/f(x)   ,  then both   f(x)  and  g(x)  will be termed as associates and in the case   f(x) = c.g(x)   for some  c is not equivalent to zero that is element of  F  .

Unit of F[x]  :-    A unit of   F[x]   is that element which have multiplicative inverse and we have shown above that in the polynomials   F[x]   the only   (multiplicatively) elements are constant polynomials .

       Hence all constant polynomials in

                                                   F[x]  are units of   F[x]  .

Subfields and field extensions

A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subset containing 0 and 1, closed under the operations +, −, · and multiplicative inverses and with its own operations defined by restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers and the rational numbers are examples.
The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A field extension F / E is simply a field F and a subfield E ⊂ F. Constructing such a field extension F / E can be done by "adding new elements" or adjoining elements to the field E. For example, given a field E, the set F = E(X) of rational functions, i.e., equivalence classes of expressions of the kind

where p(X) and q(X) are polynomials with coefficients in E, and q is not the zero polynomial, forms a field. This is the simplest example of a transcendental extension of E. It also is an example of a domain (the ring of polynomials in this case) being embedded into its field of fractions .
The ring of formal power series is also a domain, and again the (equivalence classes of) fractions of the form p(X)/ q(X) where p and q are elements of form the field of fractions for . This field is actually the ring of Laurent series over the field E, denoted .
In the above two cases, the added symbol X and its powers did not interact with elements of E. It is possible however that the adjoined symbol may interact with E. This idea will be illustrated by adjoining an element to the field of real numbers R. As explained above, C is an extension of R. C can be obtained from R by adjoining the imaginary symbol i which satisfies i² = −1. The result is that R[i]=C. This is different from adjoining the symbol X to R, because in that case, the powers of X are all distinct objects, but here, i²=−1 is actually an element of R.
Another way to view this last example is to note that i is a zero of the polynomial p(X) = X² + 1. The quotient ring can be mapped onto C using the map . Since the ideal (X²+1) is generated by a polynomial irreducible over R, the ideal is maximal, hence the quotient ring is a field. This nonzero ring map from the quotient to C is necessarily an isomorphism of rings.
The above construction generalises to any irreducible polynomial in the polynomial ring E[X], i.e., a polynomial p(X) that cannot be written as a product of non-constant polynomials. The quotient ring F = E[X] / (p(X)), is again a field.

There are many topics of Field and Subfield for discussion . If anyone wants to discuss more about Field and Subfield  may post their comments for a good discussion .