I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields...

It goes "Define a $\mathbb{F}_{5}$-vector space homomorphism $\varphi$ : $Map(\mathbb{F}_{5}, \mathbb{F}_{5})$ such that $Im(\varphi)$ = $Map_{even}(\mathbb{F}_{5} , \mathbb{F}_{5})$" .

  • $\begingroup$ What do you mean by $\operatorname{Map}(\Bbb{F}^5,\Bbb{F}^5)$? Are these all maps from $\Bbb{F}^5$ to $\Bbb{F}^5$, or only group homomorphisms, or vector space homomorphisms, or field homomorphisms? And what do you mean by $\operatorname{Map}_{\operatorname{even}}(\Bbb{F}^5,\Bbb{F}^5)$? Are these morphisms $f:\ \Bbb{F}^5\ \longrightarrow\ \Bbb{F}^5$ such that $f(-x)=f(x)$? $\endgroup$ – Servaes May 24 '16 at 17:30
  • $\begingroup$ @Servaes I'm sorry I edited the question I should have written 5 as a supscript not as an exponent $\endgroup$ – Msmat May 24 '16 at 17:33
  • $\begingroup$ Oh I hadn't even noticed, I just copied what your wrote, but you are right. My questions still stand though... $\endgroup$ – Servaes May 24 '16 at 17:49
  • $\begingroup$ @Servaes ah okay. And yes by $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$ the function that maps $\mathbb{F}_{5} \rightarrow \mathbb{F}_{5}$ is meant $\endgroup$ – Msmat May 24 '16 at 18:21
  • $\begingroup$ So $\varphi$ is supposed to be a mapping from $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$ to $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$ or ...what? $\endgroup$ – Jyrki Lahtonen May 25 '16 at 3:55

For a map $f:\ \Bbb{F}^5\ \longrightarrow\ \Bbb{F}^5$ to be even means that $f(-x)=f(x)$ for all $x\in\Bbb{F}^5$. In other words, replacing $x$ by $-x$ in this function leaves its value unchanged. For example, the functions $$f(x)=x^2\qquad\text{ or }\qquad f(x)=g(x)g(-x),$$ work, where $g:\ \Bbb{F}^5\ \longrightarrow\ \Bbb{F}^5$ can be any function. Can you think of more such functions?

Now to construct such a morphism $\varphi$, you need to construct some $\varphi(f)\in\operatorname{Map}_{\operatorname{even}}(\Bbb{F}^5,\Bbb{F}^5)$ from every $f\in\operatorname{Map}(\Bbb{F}^5,\Bbb{F}^5)$, in such a way that $$\varphi(\lambda f)=\lambda\varphi(f)\qquad\text{ and }\qquad\varphi(f+g)=\varphi(f)+\varphi(g),$$ holds for all $\lambda\in\Bbb{F}^5$ and all $f,g\in\operatorname{Map}(\Bbb{F}^5,\Bbb{F}^5)$. And moreover, you want $\varphi$ to be surjective, so every even function should be constructible from some $f\in\operatorname{Map}(\Bbb{F}^5,\Bbb{F}^5)$ in this way.

The surjectivity is not something to worry about at first; once you see what kinds of constructions work to get even functions, it should not be hard to find one that makes $\varphi$ surjective. So my advice is to go wrestle with how you can construct even functions from general functions.

  • $\begingroup$ Ah I see e.g $f(x) = x^{4} , f(x) = x^{6}$ and so on ( all x with an even exponent ) am I right? Thank you very much, I think now I'm closer to get the hang of it than before. But what about the field $\mathbb{F}_{5}$ and it's elements , I have to consider here? $\endgroup$ – Msmat May 24 '16 at 18:53

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