# irreducible polynomial

• Jun 4th 2010, 03:41 AM
Aki
irreducible polynomial
Is the following statement true ? If so, how is it proven ?

(1) Let
$\displaystyle f(x_1,\cdots,x_n) \in K[x_1,\cdots,x_n]$
is irreducible, where $\displaystyle K$ is a field.
Then it is still irreducible after any variable change
$\displaystyle x_i=\phi_i(y_1,\cdots,y_m), \; i=1,\cdots,n.$

(2) $\displaystyle K[x_1,\cdots,x_n]$ is not a principal ideal domain,
when $\displaystyle n \neq 1$.

• Jun 4th 2010, 04:40 AM
Swlabr
Quote:

Originally Posted by Aki
Is the following statement true ? If so, how is it proven ?

(1) Let
$\displaystyle f(x_1,\cdots,x_n) \in K[x_1,\cdots,x_n]$
is irreducible, where $\displaystyle K$ is a field.
Then it is still irreducible after any variable change
$\displaystyle x_i=\phi_i(y_1,\cdots,y_m), \; i=1,\cdots,n.$

(2) $\displaystyle K[x_1,\cdots,x_n]$ is not a principal ideal domain,
when $\displaystyle n \neq 1$.

For (2), assume $\displaystyle n \geq 2$ then look at the ideal generated by $\displaystyle x_1$ & $\displaystyle x_2$.