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Let $K$ be a Galois extension of $F$, and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$, and $H=\mathrm{Gal}(K/F(a))$. Let $z_1, z_2,\ldots,z_r$ be left coset representatives of $H$ in $G$. Show that $$\min(F,a)=\prod_{i=1}^r\left( x - z_i(a)\right).$$

In this product $\min(F,a)$ is of degree $r$. That is true from fundamental theorem.

And if one of $z_i$ is the identity, then $a$ satisfies the polynomial.

My question is:

What is the guarantee that $z_i(a)$ belongs to $F$ for all $i$?

What if I choose a representative which is not the identity of $\mathrm{Gal}(K/F)$?

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    $z_i(a)$ is not in $F$, unless $a$ is already in $F$.2012-02-16

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