# Can any one beat this?(Polynomials)

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• Nov 25th 2009, 01:24 AM
Xingyuan
Can any one beat this?(Polynomials)
If $g.c.d(f(x),g(x))=1$,

where $f(x) \in \mathbb{K}[x]$ and $g(x) \in \mathbb{K}[x]$,

where $\mathbb{K}$ is a field of number.

for $\forall$ $m \in \mathbb{Z}$ and $m>0$

show that:

$g.c.d(f(x^m),g(x^m))=1$
(Giggle)
• Nov 25th 2009, 03:15 AM
Shanks
If gcd (f(x), g(x))=1, then there exist u(x) and v(x) in K[x] such that
u(x)f(x)+v(x)g(x)=1,
Thus for any integer m>0,
u(x^m)f(x^m)+v(x^m)g(x^m)=1
gcd(f(x^m),g(x^m))=1