I have a linear map from [itex]$ V\rightarrow K[X_{1},...,X_{n}]\rightarrow K[X_{1},...,X_{n}]/I.$[/itex](adsbygoogle = window.adsbygoogle || []).push({});

how do i prove that a linear map from [itex]$ V=\{$[/itex]polynomials with [itex]$\deg _{x_{i}}f\prec q\}$[/itex] to [itex]$ K[X_{1},..X_{n}]/I.$[/itex] where I is the ideal generated by the elements [itex]$ X_{i}^{q}-X_{i},1\leq i\leq n.,$[/itex] is both surjective and that the kernel is zero. V is a vector space over K. Have [itex]$ dim_{k}V=\{$[/itex]the number of different monomials\}= [itex]$ q^{n}.$[/itex] and [itex]$ \mid V\mid =q^{q^{n}}.$[/itex] K is a field with q elements.

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# Linear transformation, show surjection and ker=0.

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