Field with 729 elements.

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.

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

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 $\;\Bbb F_p\;$