Mg Problem 33

33.

a. Prove that if $[0] \neq [a] \in \mathbb{F}_{p}$, then $\{[a],[2a], ... ,[(p-1)a]\}= \mathbb{F}_{p}^{\times}$

Since $p$ is prime, no multiple of $a$ has $p$ as a factor until $pa$. Therefore, every multiple of $a$ from $a$ to $(p-1)a$ is not a member of the equivalence class of $p$.

b.

c.

d. Prove that $\mathbb{Q}[x]$ is isomorphic to the ring of polynomial functions $Func_{\mathbb{Q}[x]}$ by showing injectivity of the natural map from $\mathbb{Q}[x]$ into $Func_{\mathbb{Q}[x]}$.

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