Exercise 7.7

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Generalize Exercise 6 by showing that if $\alpha$ is not a square in $F$, it is not a square in any extension of odd degree and is a square in every extension of even degree.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
Write $n = [K:F]$, and $q = \mathrm{Card}\ F$.

As $\alpha$ is not a square in $F$, the characteristic of $F$ is not 2 (see Ex.7.6), and $\alpha^{(q-1)/2} 
e 1$. Since $\alpha^{q-1} = 1$, $\alpha^{(q-1)/2}= -1$.
$$\alpha^{(q^n-1)/2} =( \alpha^{(q-1)/2})^{1+q+\cdots+q^{n-1}} = (-1)^{1+q+\cdots+q^{n-1}} .$$

$\bullet$ If $n$ is odd, $1 + q +\cdots+q^{n-1} \equiv 1 \pmod 2$, thus $\alpha^{(q^n-1)/2} = -1
e 1$, and consequently $\alpha$ is not a square in $K$.

$\bullet$ If $n$ is even, as $q$ is odd ($\mathrm{char}(F) 
e 2$), $1 + q +\cdots+q^{n-1} \equiv 0 \pmod 2$, thus $\alpha^{(q^n-1)/2} = 1$, so $\alpha$ is a square in $K$.
\end{proof}

Exercise 7.7

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Exercise 7.7

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