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I don't understand the hint. The point is that $$\Bbb{F}_{q^2}=\Bbb{F}_p(\zeta_{q+1})$$ (proof: write $q=p^n$, then $\Bbb{F}_p(\zeta_{q+1})=\Bbb{F}_{p^m}$ where $m$ is the least integer such that $p^... • 69.2k 2 votes ### Summation involving addition modulo$2$Suppose$w\not\in S^\perp$. We show that$\sum_{s\in S} (-1)^{s\cdot w} = 0$by finding for every$s \in S$such that$s \cdot w = 1$an$r\in S$such that$r\cdot w = 0$. Thus by pairing the sum is ... • 706 2 votes ### Subfield generated by traces of some elements in finite fields I might do this as follows, related to the hint in a way. Turning it into a counting argument. Consider the set $$S=\{z\in\Bbb{F}_{q^2}\mid z^{q+1}=1\}.$$ As$\Bbb{F}_{q^2}^*$is cyclic of order$q^2-...
To summarize the answers in the comments, gcd of polynomials is defined as the greatest common divisor $g$ of $a$ and $b$ where $deg(g) \geq deg(h)$ where $h$ is any other common divisor of $a$ and $b$...