I was working on this problem from an old qual exam and here is the
question. In particular this is not for homework.
True or False: There are no fields of order 32. Justify your answer.
Attempt: From general theory I know that any finite field has prime power order
and conversely given any prime power there exists a finite field of that order. So of course such fields exist. But now I need to explicitly construct such a field.
If I could somehow construct $\mathbb{Z}_2[x]/(p(x))$ where $p(x)$ is a polynomial of degree 5 which is irreducible over $Z_2$ I am done. But wait, how do I come up with a degree 5 polynomial that is irreducible over $Z_2$. My normal methods don't work here because $p(x)$ does not have order 2 or 3. In which case it is easy to check for irreducibility.
My question is in these kinds of situations, is there a general way to proceed.
Note: I have not learnt Galois' theory or anything like that. Does this problem require more machinery to solve?
Please help.
Best Answer
No more machinery. Make yourself a table of irreducible polynomials of degrees up to 5 by thinking about how to recognize polynomials over $\mathbb Z_2$ with $0$ or $1$ as a root, then proceeding by doing a sieve of Eratosthenes (crossing out polynomials that are divisible by lower-degree irreducibles).