Showing subfields using characteristic

[jacobsldr]

The question: Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}.

I know since it is a field that it is also an integral domain and by theorem the characteristic is either 0 or prime. By LaGranges I believe I can say that all subfields would have to have order that divides 32. This would leave me to say that the char F is either 0 or 2.

F will be a subfield of itself by definition. I am just not sure how to put the logic to say that {0,1} is the only other subfield.

Any hints would be appreciated. Thanks.

 

[DrMike]

The question: Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}.

I know since it is a field that it is also an integral domain and by theorem the characteristic is either 0 or prime. By LaGranges I believe I can say that all subfields would have to have order that divides 32. This would leave me to say that the char F is either 0 or 2.

F will be a subfield of itself by definition. I am just not sure how to put the logic to say that {0,1} is the only other subfield.

Any hints would be appreciated. Thanks.

One step closer :

If the characteristic is zero, then 1+1+..+1 is never 0. However, you have a finite field....

After that, I'd suggest using the fact that your field must be an extension field of Z[sub:2pz3ug9k]2[/sub:2pz3ug9k] by some polynomial root, noting that 0 and 1 must be in any subfield, and analysing what happens if the subfield contains another element x.