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Thread: irreducible polynomial

  1. #1
    Aki
    Aki is offline
    Junior Member Aki's Avatar
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    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 $\displaystyle n \neq 1$.

    Thank you in advance.
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  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
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    Quote Originally Posted by Aki View Post
    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 $\displaystyle n \neq 1$.

    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 $\displaystyle n \geq 2$ then look at the ideal generated by $\displaystyle x_1$ & $\displaystyle x_2$.
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