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Unformatted text preview: 426 9. FIELD EXTENSIONS – SECOND LOOK The formal derivative in KŒx is deﬁned by the usual rule P calcufrom
lus: We deﬁne D.x n / D nx n 1 and extend linearly. Thus, D. kn x n / D
nkn x n 1 . The formal derivative satisﬁes the usual rules for differentiation:
9.3.1. Show that D.f .x/Cg.x// D Df .x/CDg.x/ and D.f .x/g.x// D
D.f .x//g.x/ C f .x/D.g.x//.
(b) Suppose that the ﬁeld K is of characteristic zero. Show that
Df .x/ D 0 if, and only if, f .x/ is a constant polynomial.
Suppose that the ﬁeld has characteristic p . Show that Df .x/ D 0
if, and only if, there is a polynomial g.x/ such that f .x/ D
g.x p /. 9.3.3. Suppose f .x/ 2 KŒx, L is an extension ﬁeld of K , and f .x/
factors as f .x/ D .x a/g.x/ in LŒx. Show that the following are
(c) a is a multiple root of f .x/.
g.a/ D 0.
Df .a/ D 0. 9.3.4. Let K Â L be a ﬁeld extension and let f .x/; g.x/ 2 KŒx. Show
that the greatest common divisor of f .x/ and g.x/ in LŒx is the same
as the greatest common divisor in KŒx. Hint: Review the algorithm for
computing the g.c.d., using division with remainder.
9.3.5. Suppose f .x/ 2 KŒx, L is an extension ﬁeld of K , and a is a
multiple root of f .x/ in L. Show that if Df .x/ is not identically zero,
then f .x/ and Df .x/ have a common factor of positive degree in LŒx
and, therefore, by the previous exercise, also in KŒx.
9.3.6. Suppose that K is a ﬁeld and f .x/ 2 KŒx is irreducible.
(b) Show that if f has a multiple root in some ﬁeld extension, then
Df .x/ D 0.
Show that if Char.K/ D 0, then f .x/ has only simple roots in
any ﬁeld extension. The preceding exercises establish the following theorem:
Theorem 9.3.1. If the characteristic of a ﬁeld K is zero, then any irreducible polynomial in KŒx has only simple roots in any ﬁeld extension. ...
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