# Question concerning decomposition

• Oct 11th 2013, 10:32 AM
huberscher
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})\$?
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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