Galois group of $prod_{i=1}^{p-1} (X^2-i)$
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Let $p$ be an odd prime. How do I compute the Galois group of $F/Z_p$ where $F$ is the splitting field of $prod_{i=1}^{p-1} (X^2-i)$ over $Z_p$? Since exactly $frac{p-1}{2}$ integers from $1leq ileq p-1$ are the roots of $X^2-i$ in $Z_p$, let's write $F=Z_p(sqrt{i_1},...,sqrt{i_k})$ where $k=frac{p-1}{2}$ and $sqrt{i_j} notin Z_p$. This leads me to guess that the Galois group is $(Z_2)^k$, but is it really true? How do I prove this?
abstract-algebra field-theory galois-theory
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edited Jan 9 at 20:04
Jyrki Lahtonen
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