Let $f(x) = x^4 + 6x^3 + 15x^2 + 10x + 1$ and $g(x) = 2x^2 + 15x + 1$.
Consider $f$ and $g$ as polynomials with coefficients in (a) $\mathbb Q$, (b) $\mathbb F_2$, (c) $\mathbb F_3$, and (d) $\mathbb F_5$. Answer the following question in each case.
Determine the degree of $f(x)$ and $g(x)$.
My attempt:
(a) In $\mathbb Q$, Degree of $f$ is $4$ and Degree of $g$ is $2$
(b) After reducing $\pmod 2$ Would the functions become: $f(x) = x^4 + 0x^3 + 1x^2 + 0x + 1=x^4 + x^2 + 1 $ and $g(x) = 0x^2 + 1x + 1 = x+1$ , so degree of $f$ is $4$ and of $g$ is $1$
Am I doing the right thing? Any help is appreciated.
Yes, what you are doing is correct.
Furthermore, note that mod $3$ and mod $5$ the leading coefficients of $f$ and $g$ are not canceled. So over $\mathbb F_3$ and $\mathbb F_5$, the degrees coincide with the degrees over $\mathbb Q$.