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Unformatted text preview: 425 9.3. THE DERIVATIVE AND MULTIPLE ROOTS Q
Corollary 9.2.5. Let p.x/ 2 KŒx, and suppose L and L are two splitting
ﬁelds for p.x/. Then there is an isomorphism W L ! L such that
.k/ D k for all k 2 K . Q
Proof. Take K D K and D id in the proposition. I Exercises 9.2
9.2.1. For any polynomial f .x/ 2 KŒx there is an extension ﬁeld L of K
such that f .x/ factors into linear factors in LŒx. Give a proof by induction
on the degree of f .
9.2.2. Verify the following statements: The rational polynomial f .x/ D
x 6 3 is irreducible over Q. It factors over Q.31=6 / as
If ! D e
.x 31=6 /.x C 31=6 /.x 2
i=3 , then the irreducible factorization of f .x/ over Q.31=6 ; !/ 31=6 /.x C 31=6 /.x 9.2.3.
(c) 31=6 x C 31=3 /.x 2 C 31=6 x C 31=3 /: ! 31=6 /.x ! 2 31=6 /.x ! 4 31=6 /.x ! 5 31=6 /: Show that if K is a ﬁnite ﬁeld, then KŒx has irreducible elements
of arbitrarily large degree. Hint: Use the existence of inﬁnitely
many irreducibles in KŒx, Proposition 1.8.9.
Show that if K is a ﬁnite ﬁeld, then K admits ﬁeld extensions of
arbitrarily large ﬁnite degree.
Show that a ﬁnite–dimensional extension of a ﬁnite ﬁeld is a ﬁnite ﬁeld. 9.3. The Derivative and Multiple Roots
In this section, we examine, by means of exercises, multiple roots of polynomials and their relation to the formal derivative.
We say that a polynomial f .x/ 2 KŒx has a root a with multiplicity
m in an extension ﬁeld L if f .x/ D .x a/m g.x/ in LŒx, and g.a/ ¤ 0.
A root of multiplicity greater than 1 is called a multiple root. A root of
multiplicity 1 is called a simple root. ...
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