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Let $\mathbb{F}$ be a field with 729 elements. How many distinct proper subfields does $\mathbb{F}$ contain. Please be generous and tell the reason also.

Thanks.

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  • $\begingroup$ Can you think of any subfields? $\endgroup$ – curious Nov 5 '14 at 17:13
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    $\begingroup$ What do you know about the structure of finite fields which might helping answering this question? $\endgroup$ – Mark Bennet Nov 5 '14 at 17:13
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    $\begingroup$ You've got some good hints here already. One more hint: $729=3^{6}$. $\endgroup$ – Alex Wertheim Nov 5 '14 at 17:14
  • $\begingroup$ ans is 3 right? $\endgroup$ – user180150 Nov 5 '14 at 17:30
  • $\begingroup$ Yes, it is @देवेन्द्रprasad : $\;\Bbb F_3\;,\;\;\Bbb F_{3^2}\;,\;\;\Bbb F_{3^3}\;$ $\endgroup$ – Timbuc Nov 5 '14 at 20:38
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Hint: a field $\;\Bbb F_{p^m}\;$ is a subfield of $\;\Bbb F_{p^n}\;$ iff $\;m\mid n\;$ .

Further hint: in order to prove the above , it may be really helpful to consider all those fields as linear spaces over their common prime field $\;\Bbb F_p\;$

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