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Math Help - 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
    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.
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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
    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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