I am reading a paper, and have a next expresion Let $\sigma(X) \in F_{2^m}[X]$ with deg $(\sigma(X)) = \delta$, then writing $\sigma_1$ for the even part of $\sigma$ and $X\sigma'$ for the odd part of $\sigma$, obtain: $\sigma(X) = \sigma_1(X)+X\sigma'(X)$,
My question: How to prove this?
pdta: I know that $X$ is a odd function but I don't know $\sigma'(X)$, then I am not sure whether $X\sigma'(X)$ is odd or even function, ...