1. ## Question concerning decomposition

Hello

two questions:
ç
a) Write $\displaystyle \mathbb{F}_9[y] \diagup (1+y+y^3+y^4)$ as a direct sum.
b) Why doesn't exist a direct sum decomposition of $\displaystyle \mathbb{F}_{5^{2744}}[y] \diagup (y^{160675})$?
ç

a) Since $\displaystyle f(y)=1+y+y^3+y^4 = (1+y)*(1+y^3)=h(y)*g(y)$ the roots of f are contained in the field considered. y is a root of f if it's a root of h or it is a root of g. So my decomposition here is

$\displaystyle \mathbb{F}_9[y] \diagup (1+y) \oplus \mathbb{F}_9[y] \diagup (1+y^3)$

Is it correct and what about my reason? 9=3^2 but why can that not be decomposed more?

b) $\displaystyle 2744=2^3*7^3$
$\displaystyle 160675=5^2*6427$

Are the numbers choosed with this factorisation for causing confusion only? My reason here is:

If y is a root of $\displaystyle f(y)=y^{160675}$ it is not a root of a polynomial dividing f(y) of a lower degree. No matter what $\displaystyle y^{160674}$ is the ys contained in the field considered at b) are only 0 when powered to 160675.

What do you think of my explanations? How would you reason here?

Regards