Normal? Well, I'm going to assume that you can extend your arms and things, and that you're finite. Thus, normal would be:

Suppose L is a field extension of a field K. In other words, K is a subfield of L. L is termed a finite normal extension of K if it is both a finite extension and a normal extension of K. More explicitly, it satisfies the following equivalent conditions:

    L is a finite extension of K, and if p(x) \in K[x] is an irreducible polynomial having a root in L, then p splits completely into linear factors over L[x].
    L is a finite extension of K, and the minimal polynomial of any element of L splits completely over K.
    L is the splitting field over K of a finite collection of irreducible polynomials.
    L is the splitting field over K of a single polynomial (not necessarily irreducible).
    L is the splitting field over K of a single irreducible polynomial.

I hope that this makes it clear.