This is Exercise 16 from page page 83 of Analysis I from Amann and Escher.
Exercise:
I have some further context at the bottom of this post.
My attempt:
I don't understand the hint at all. On an earlier page he defines $\mathbb F_2 := \big( \{ 0, 1 \}, +, \cdot \big)$. I checked the index and I don't see a definition of $\mathbb F_2 [X]$. Is it a polynomial field with two elements? Or a polynomial field containing only polynomials of order 2? I'm not sure the latter makes sense, because it wouldn't be closed under multiplication.
I understand that to prove injectivity, you want to show that two outputs being equal implies the inputs are equal, or equivalently that different inputs implies different outputs. But the hint is an element of the domain, right?
Another question I have is why the proof of Remark 8.14 (c) doesn't apply in the case of this exercise. (The Remark and context are at the bottom of this post.) A field is just a nice ring, and the proof below applies to a nontrivial finite ring, which as far as I can tell is applicable to the exercise. Can't I just point to that proof given by the book and be done with it?
I appreciate any help.
Context:
