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How to prove that every finite field of order $125$ has a subfield of order $25$. In general what is the strategy to attack such kind of problems?

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    Consider objects of the form $x^5$, where $x$ ranges over the big field.2012-11-10

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Consider a field $F$ with $q$ elements, $q$ being a power of some prime. Suppose $L$ is a field containing $F$, with $[L\colon F]=m$. Since $L$ is an $F$-vector space of dimension $m$, $|L|=q^m$. Thus $[\mathbb F_{125}\colon\mathbb F_5]=3$. Now do you see why there’s no field strictly between these two fields?

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    You don't need to consider $\Bbb F_5$, simply $[\Bbb F_{125}:\Bbb F_{25}]$ would have to be $3/2$, which is absurd.2012-11-10