Graphics Reference
In-Depth Information
B.8.10. Theorem.
Let k be a field. A nonconstant polynomial f(X) Πk[X] has a mul-
tiple factor g(X) if and only if g(X) is also a factor of f ¢(X).
Let f(X) = g(X)
n
h(X). The product rule of the derivative easily leads to the
Proof.
result.
B.8.11. Theorem.
If k is a field, then k[X] is a PID (and hence a UFD).
Proof.
See [Mill58].
Only the polynomial ring in
one
variable over a field is a PID. To see that
k[X
1
,X
2
,...,X
n
] is not a PID if n > 1, one simply needs to convince oneself that the
ideal in k[X,Y] consisting of all polynomials whose constant term is 0 is not a prin-
cipal ideal.
Definition.
Let f
1
, f
2
,..., f
k
Πk[X]. A
greatest common divisor
of the f
i
, denoted by
gcd(f
1
,f
2
,...,f
k
), is a largest degree polynomial g Πk[X] that divides each of the f
i
. A
least common multiple
of the f
i
, denoted by lcm(f
1
,f
2
,...,f
k
), is a smallest degree poly-
nomial h Πk[X] with the property that each f
i
divides h.
B.8.12. Theorem.
Let f
1
, f
2
,..., f
k
Πk[X].
(1) The polynomials gcd(f
1
,f
2
,...,f
k
) and lcm(f
1
,f
2
,...,f
k
) are unique up to scalar
multiple.
(2) <f
1
,f
2
,...,f
k
> = <gcd(f
1
,f
2
,...,f
k
)>.
Proof.
This is easy. The key fact is that k[X] is a UFD.
B.8.13. Theorem.
If k is a field, then both k[[X
1
,X
2
,...,X
n
]] and k[X
1
,X
2
,...,X
n
]
are Noetherian.
Proof.
First note that k satisfies the ascending chain condition since it has only two
ideals, namely, 0 and itself. Now use Corollary B.7.10 and Theorem B7.11.
Definition.
Let K be a field. If k is a subfield of K, then K is called an
extension
(field)
of k.
Definition.
Let K
1
and K
2
be extension fields of a field k. An isomorphism
s :K
1
Æ K
2
is called an
isomorphism over k
if s is the identity on k. If K = K
1
= K
2
,
then s is called an
automorphism of K over k
.
Definition.
Let k be a subfield of a field K. If A Õ K, then k(A) will denote the
small-
est subfield of K containing k and A
, more precisely,
()
=«
{
}
.
k A
F F is a subfield of K which contains k and A
