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.

  • $\begingroup$ As a minor comment, those are fields (not groups). $\endgroup$
    – Qudit
    Commented Feb 27, 2015 at 3:56
  • $\begingroup$ Yes you are. reducing is exactly reducing the coefficients. And as usual, you check the degree after the reduction is done. $\endgroup$ Commented Feb 27, 2015 at 10:46

1 Answer 1


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 $\mathbb Q$.


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