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It's not clear to me what rational function fields are and the significance of normal extensions in the first place.

$Z_7(t^9)$ seems smaller than $Z_7(t)$ so it makes sense one would be contained in the other but that is where I am stuck

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I suppose the first step is to look up the definitions of rational function fields and of normal extensions. –  Chris Eagle Oct 24 '12 at 9:21
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