The characteristic of a ring is usually defined as the smallest positive integer such that adding times to itself gives . If no such number exists, we say the characteristic is 0. Another way to define it is as the smallest positive integer such that adding to itself times produces , for any . The two definitions are equivalent, can you prove it?
Now, onto your question! IF is a field of characteristic , then either or is prime. (You can prove this)
It's not hard at all to see that a field and any of its extensions must have the same characteristic. They share a multiplicative identity.
As a next step, you should take the definitions of the things you are using and play around with them to see if you can make any progress.