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!

  • 2
    $\begingroup$ That doesn't look like a field, since $y^{32134}\cdot y=0$ but a field has no zero divisors. $\endgroup$ – hmakholm left over Monica Oct 11 '13 at 18:52
  • $\begingroup$ Because every field is an integral domain and this means it has no proper zero devisors? $\endgroup$ – Lullaby Oct 11 '13 at 18:56
  • $\begingroup$ I guess the problem statement is just a slurring of "for the finite field $F_{5^{392}}$, this other quotient ring can be decomposed into a sum...." $\endgroup$ – rschwieb Oct 11 '13 at 19:39
  • $\begingroup$ @Lullaby: That's one way of putting it. You can also see directly that a proper zero divisor can't be invertible (and therefore can't be a field element): If $ab=0$ and $a$ is invertible, then $b=1b=a^{-1}ab=a^{-1}0=0$. $\endgroup$ – hmakholm left over Monica Oct 11 '13 at 20:06
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    $\begingroup$ Henning: The question is whether this is a direct sum of fields - not whether it is a field. For example $F_5[y]/(y^2-1)$ is not a field, but it is a direct sum of the fields $F_5[y]/(y-1)$ and $F_5[y]/(y+1)$ by the Chinese Remaider Theorem. Note that $F_5[y]/(y^2-1)$ has zero divisors corresponding to a factorization of $y^2-1$. But, as rschwieb observed, a direct sum of fields does not have nilpotent elements. $\endgroup$ – Jyrki Lahtonen Oct 12 '13 at 4:47

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 $y$.


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