I am stuck in a 'true or false' question about decomposition into direct sum of finite fields and don't really know how to prove the problem. Can anybody give me a hint or an idea how to solve it, please?
We have the field with finitely many elements $F_{5^{392}}[y]/(y^{32135})$. It can be decomposed into a direct sum of finite fields. True or false?
$32135$ is not a prime number, its prime factorization is $5 \times 6427$, but this is not relevant to the proof, i guess. 🙁
Can anybody help me? Thank you in advance!
Best Answer
The ring $F_{5^{392}}[y]/(y^{32135})$ is a $32135$ dimensional $F_{5^{392}}$ vector space, which means it is vector space isomorphic to the direct sum of $32135$ copies of $F_{5^{392}}$.
You can arrive at this conclusion by thinking of the quotient as the vector space of polynomials in that field with degree strictly less than $32135$. This should assist you in thinking about a basis. This is a direct sum of vector spaces, not rings.
The ring is not ring isomorphic to a direct product of fields (as rings, not vector spaces). This is because a direct product of fields has no nonzero nilpotent elements, and yet this quotient definitely has nonzero nilpotent elements, namely $y$.