irreducible polynomial

**Aki** · June 4th 2010, 03:41 AM

Is the following statement true ? If so, how is it proven ?

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

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

Thank you in advance.

**Swlabr** · June 4th 2010, 04:40 AM

Originally Posted by Aki

Is the following statement true ? If so, how is it proven ?

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

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

Thank you in advance.

For (1), notice that a variable change is an automorphism of your ring. This means that it has an inverse...

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

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