Problem: Describe the ring structure of F_2[x]/(x^4 + x^2 + 1).
Quotient rings are very difficult for me. What are the elements of this quotient ring? Is this polynomial irreducible? How do I determine this? Do I try to divide out by the degree 1 polynomials in this quotient ring? If the poly is irreducible, then this is a field, right? How do the elements in this ring differ from F_2[x]/(x^4 + x + 1)?
Sorry for being so stupid! :<
February 22nd 2009, 11:46 PM
has no root in so it can't be written as a product of irreducible polynomials of degree 1 and 3. The other and last possibility for it to be reducible is to be the product of two irreducible polynomials of degree 2. The only irreducible polynomial of degree 2 in is (exercice), and we have
Therefore is reducible, and:
( )(wrong) which is not a field, since it's not an integral domain.
With the same idea of proof, you can show that is irreducible in and then, since this ring is a principal ideal domain, is a field.
and are fields, and
By unicity up to isomorphism for finite fields, and
In conclusion, ( )(wrong) while
February 22nd 2009, 11:50 PM
Thank you so much again clic-clac! This is by far the most helpful forum I've found; I'm donating to it right away.
I'm sorry I can't write that like that. When is a field and are two polynomials in if and are relatively prime, then
But of course, and itself are not relatively prime polynomials.
What remains true is that is not an integral domain: is such that but (indeed ).
One more thing: to see that the "isomorphism" quoted is wrong, one may note that the first ring has a non-zero nilpotent element, , while the second, as a product of fields, has no non-zero nilpotent element.