Separable Extension - Informal Discussion

Informal Discussion

The reader may wish to assume that, in what follows, F is the field of rational, real or complex numbers, unless otherwise stated.

An arbitrary polynomial f with coefficients in some field F is said to have distinct roots if and only if it has deg(f) roots in some extension field . For instance, the polynomial g(X)=X2+1 with real coefficients has precisely deg(g)=2 roots in the complex plane; namely the imaginary unit i, and its additive inverse −i, and hence does have distinct roots. On the other hand, the polynomial h(X)=(X−2)2 with real coefficients does not have distinct roots; only 2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg(f)=2 roots.

In general, it can be shown that the polynomial f with coefficients in F has distinct roots if and only if for any extension field, and any, does not divide f in E. For instance, in the above paragraph, one observes that g has distinct roots and indeed g(X)=(X+i)(X−i) in the complex plane (and hence cannot have any factor of the form for any in the complex plane). On the other hand, h does not have distinct roots and indeed, h(X)=(X−2)2 in the complex plane (and hence does have a factor of the form for ).

Although an arbitrary polynomial with rational or real coefficients may not have distinct roots, it is natural to ask at this stage whether or not there exists an irreducible polynomial with rational or real coefficients that does not have distinct roots. The polynomial h(X)=(X−2)2 does not have distinct roots but it is not irreducible as it has a non-trivial factor (X−2). In fact, it is true that there is no irreducible polynomial with rational or real coefficients that does not have distinct roots; in the language of field theory, every algebraic extension of or is separable and hence both of these fields are perfect.