daon

05-16-2007, 02:42 PM

This was on my final, and I couldn't get it entriely.

Prove that \L \,\, \,\, \bigcup_{k \ge 0} GF\(p^k\) \,\, \,\, is a field.

So many thoughts ran though my head. 1) Induct on k? 2) Try a proof by contradiction? 3) Show directly that it is a unitary, commutative ring with all non-zero elements being units?

This idea also came to me, but I felt like I was digging myself into a hole:

If d | n then GF(p^d) is a subfield of GF(p^n). And hence all fields GF(p^k) with k<n would be subfields of GF(p^{n!}) and hence there is always some "larger finite field" which contains all other fields in the union chain. I'm not sure where to go from here with this.

After spending a few minutes on each possibility, I gave up. Any ideas?

Prove that \L \,\, \,\, \bigcup_{k \ge 0} GF\(p^k\) \,\, \,\, is a field.

So many thoughts ran though my head. 1) Induct on k? 2) Try a proof by contradiction? 3) Show directly that it is a unitary, commutative ring with all non-zero elements being units?

This idea also came to me, but I felt like I was digging myself into a hole:

If d | n then GF(p^d) is a subfield of GF(p^n). And hence all fields GF(p^k) with k<n would be subfields of GF(p^{n!}) and hence there is always some "larger finite field" which contains all other fields in the union chain. I'm not sure where to go from here with this.

After spending a few minutes on each possibility, I gave up. Any ideas?