4 votes

Subfield generated by traces of some elements in finite fields

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^...
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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 ...
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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-...
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2 votes

How is GCD defined in polynomials with coefficients in finite fields? SageMath claims gcd(xy, x) = 1 in (GF(2^128)/p(x))[y]

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 $b$...
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