Unformatted text preview: 54 1. ALGEBRAIC THEMES factor polynomials in QŒx by exhaustive search, as there are inﬁnitely
many polynomials of each degree.
We ﬁnish this section with some elementary but important results relating roots of polynomials to divisibility.
Proposition 1.8.22. Let p 2 KŒx and a 2 K . Then there is a polynomial
q such that p.x/ D q.x/.x a/ C p.a/. Consequently, p.a/ D 0 if, and
only if, x a divides p . Proof. Write p.x/ D q.x/.x a/ C r , where the remainder r is a constant.
Substituting a for x gives p.a/ D r .
I Deﬁnition 1.8.23. Say an element ˛ 2 K is a root of a polynomial p 2
KŒx if p.˛/ D 0. Say the multiplicity of the root ˛ is k if x ˛ appears
exactly k times in the irreducible factorization of p . Corollary 1.8.24. A polynomial p 2 KŒx of degree n has at most n roots
in K , counting with multiplicities. That is, the sum of multiplicities of all
roots is at most n. Proof. If p D .x ˛1 /m1 .x ˛2 /m2 .x ˛k /mk q1 qs , where the
qi are irreducible, then evidently m1 C m2 C C mk Ä deg.p/.
I Exercises 1.8
Exercises 1.8.1 through 1.8.2 ask you to prove parts of Proposition 1.8.2.
1.8.1. Prove that addition in KŒx is commutative and associative, that 0
is an identity element for addition in KŒx, and that f C f D 0 for all
f 2 KŒx.
1.8.2. Prove that multiplication in KŒx is commutative and associative,
and that 1 is an identity element for multiplication.
1.8.3. Prove Proposition 1.8.5.
1.8.4. Prove Proposition 1.8.15. ...
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