It's just about halfway through the term and I've been able to do the first 5 homeworks without any help, however my class notes seem to be lacking substance at this point.
So I come to you for help.
2. Show that is irreducible in
my solution so far: aren't roots so it's irreducible
then from an example we did in class I need to find the number of elements in this field, then the number of elements in the cyclic group of nonzero elements? then what we did next in the example is find an element of this group of degree equal to the size of the group?
what I'd like to improve on this, I don't particularly like my proof of irreducibility since I have another question which is of degree 4 so I don't want to have to list all examples again, but it is so it's still not hard to do it exhaustively
3. Let be a finite field of characteristic , a subfield of , and a positive integer.
a) show that is a subfield of that contains
b) show that
5. Let be a prime and an irreducible polynomial of degree 2 in . if is an extension field of of order , prove that is irreducible in
This is due in about 12 hours from now, thanks for the help. I'm sorry I left it to the last minute, I tried hard to do it on my own
Even if you only get to this after the deadline I'd still appreciate someone guiding me through this as I want to know this stuff for my midterm next week too