Let $\mathbb{F}$, a field and let $f$, a monic polynomial above $\mathbb{F}[x]$ where $\deg (f)=d$. Let $\mathbb{K}$ to be the following set:$$\mathbb{K} = \left\{ g\in\mathbb{F}[x]\ |\ \deg g\lt \deg f \right\}$$ Prove that $\mathbb{K}$ is a field.
EDIT: Addition is define as one would expect, but multiplication is defined as the remainder of $g_1 \cdot g_2 = s\cdot f + r$. Meaning, $g_1 \cdot_\mathbb{K} g_2 = r$.
EDIT2: $f$ is irreducible.
It is easy to see that our "zero" is $f_0 = 0$ and our "one" is $f_1 = f+1$.
Hence, I need to prove that for every $g_1$ there's an inverse, $g_2$ such that $g_1\cdot g_2 = f+1$, but it doesn't quite working.
Also, I can rely on the fact that for every polynomial in $\mathbb{K}$ there're $s,r$ such that $g= s\cdot f + r$ where $\deg r < \deg g$.
Note: When I wrote $1$ I used the $1$ of $\mathbb{F}$ (I think that's a correct use)
What you have defined is an alternative way of looking at the quotient ring ${\mathbb F}[x]/(f)$. The map ${\mathbb K} \to {\mathbb F}[x]/(f)$, $g \mapsto g + (f)$ is a ring isomorphism. In fact, the easiest way to show that ${\mathbb K}$ is a indeed ring, is probably to show that this map is a bijection of sets and preserves the operations.
Because $f$ is irreducible, ${\mathbb F}[x]/(f)$ is a field and therefore ${\mathbb K}$ is too.