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B.8.4. Example.
The field Q of rational numbers is the quotient field of the integers
Z .
B.8.5. Theorem. Let R be a commutative ring with unity element and let I be an
ideal in R. Then I is a maximal ideal if and only if R/I is a field.
Proof.
See [Mill58].
B.8.6. Corollary.
Every maximal ideal in a commutative ring with unity element is
prime.
Proof.
This follows from Theorems B.6.3 and B.8.5.
The next theorem and its proof is the basic division algorithm for polynomials.
B.8.7. Theorem. Let k be a field. If f(X), g(X) Œ k[X] and g(X) π 0, then there exist
unique polynomials q(X), r(X) Πk[X] such that
(1) f = q g + r, and
(2) r = 0 or degree r < degree g.
Proof.
See [Dean66].
B.8.8. Theorem. Let k be a field. If f(X) Πk[X], then for any c Πk there exists a
unique polynomial q(X) Πk[X] such that
() =-
(
) (
) + ( .
fX
X cqX
fc
Proof.
See [Dean66].
An easy consequence of Theorem B.8.8 is
B.8.9. Corollary.
Let k be a field. If f(X) Πk[X], then c is a zero of f if and only if
X - c divides f.
Let k be a field and f(X) Πk[X]. If c is a root of f(X), then it follows from Corol-
lary B.8.9 that we can express f(X) in the form
n
() =-
(
)
(
) ,
fX
X c gX
where c is not a root of g(X).
Definition. The integer n is called the multiplicity of the root c of f(X). If n = 1, then
one calls c a simple root . If n > 1, then one calls c a multiple root . More generally, if
g(X) is a factor of f(X) and if n is the largest integer with the property that g(X) n
divides f(X), then n is called the multiplicity of the factor g(X). We call g(X) a multi-
ple factor if n > 1 .
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