
More about Roots 
We return to the general exposition. Let be any field and, as usual, let be the ring of polynomials in over .
DEFINITION: If in , then the derivative of written as , is the polynomial in . To make this definition or to prove the basic formal properties of the derivative, as applied to polynomials, does not require the concept of a limit. However, since the field is arbitrary, we might expect some strange things to happen. At the end of Section 3.2, we defined what is meant by characteristic of a field. Let us recall it now. A field is said to be of characteristic 0 if for in and an integer. If for some and some then is said to be of finite characteristic. In this second case, the characteristic of is defined to be the smallest positive integer such that for all It turned out that if is of finite characteristic then its characteristic is a prime number. We return to the question of the derivative. Let be a field of characteristic . In this case, the derivative of the polynomial is . Thus the usual result from the calculus that a polynomial whose derivative is must be a constant no longer need hold true. However, if the characteristic of is and if for it is indeed true that (see Problem 1). Even when the characteristic of is we can still describe the polynomials with zero derivative; if then is a polynomial in (see Problem 2). We now prove the analogs of the formal rules of differentiation that we know so well. LEMMA 5.5.1: For any and any 1. . 2. . 3. . Recall that in elementary calculus the equivalence is shown between the existence of a multiple root of a function and the simultaneous vanishing of the function and its derivative at a given point. Even in our setting, where is an arbitrary field, such an interrelation exists. LEMMA 5.5.2: The polynomial has a multiple root if and only if and have a nontrivial (that is, of positive degree) common factor. COROLLARY 1: If is irreducible, then 1. If the characteristic of is has no multiple roots. 2. If the characteristic of is has a multiple root only if it is of the form . We shall return in a moment to discuss the implications of Corollary 1 more fully. But first, for later use in Chapter 7 in our treatment of finite fields, we prove the rather special COROLLARY 2: If is a field of characteristic then the polynomial , for , has distinct roots. Corollary 1 does not rule out the possibility that in characteristic an irreducible polynomial might have multiple roots. To clinch matters, we exhibit an example where this actually happens. Let be a field of characteristic 2 and let be the field of rational functions in over . We claim that the polynomial in is irreducible over and that its roots are equal. To prove irreducibility we must show that there is no rational function in whose square is ; this is the content of Problem 4. To see that has a multiple root, notice that its derivative (the derivative is with respect to ; for being in is considered as a constant) is . Of course, the analogous example works for any prime characteristic. Now that the possibility has been seen to be an actuality, it points out a sharp difference between the case of characteristic and that of characteristic . The presence of irreducible polynomials with multiple roots in the latter case leads to many interesting, but at the same time complicating, subtleties. These require a more elaborate and sophisticated treatment which we prefer to avoid at this stage of the game. Therefore, we make the flat assumption for the rest of this chapter that all fields occurring in the text material proper are fields of characteristic . DEFINITION: The extension of is a simple extension of if for some in . In characteristic (or in properly conditioned extensions in characteristic ; see Problem 14) all finite extensions are realizable as simple extensions. This result is THEOREM 5.5.1: If is of characteristic and if , are algebraic over then there exists an element such that . A simple induction argument extends the result from 2 elements to any finite number, that is, if are algebraic over , then there is an element such that . Thus the COROLLARY: Any finite extension of a field of characteristic is a simple extension. 