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Abstract Algebra

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Moderate

Finite Field: $p^{th}$ Power Map on Rational Functions

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Let $p\ge 2$ be a prime number, let $n\ge 1$ be an integer, and let $q=p^n$.

Let $\mathbf{F}_q$ be the finite field with $q$ elements.

Let $\mathbf{F}_q(x)$ be the field of rational functions over $\mathbf{F}_q$. By definition, it is the fraction field of the polynomial ring $\mathbf{F}_q[x]$, and therefore has elements the ratios $R(x)=A(x)/B(x)$, with $A(x), B(x)\in \mathbf{F}_q[x]$ and $B(x)\not=0$.

Let $\Phi_p: \mathbf{F}_q(x)\rightarrow \mathbf{F}_q(x)$ be the $p$th power map $R(x)\mapsto R(x)^p$, for $R(x)\in \mathbf{F}_q(x)$, and let $\varphi_p$ be its restriction to $\mathbf{F}_q$ (identified with the constant rational functions), given by $a\mapsto a^p$, for $a\in \mathbf{F}_q$.

1) The map $\Phi_p$

injective.
2) The map $\Phi_p$

surjective.
3) The map $\varphi_p$

surjective.