a) Write as a direct sum.
b) Why doesn't exist a direct sum decomposition of ?
a) Since 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
Is it correct and what about my reason? 9=3^2 but why can that not be decomposed more?
Are the numbers choosed with this factorisation for causing confusion only? My reason here is:
If y is a root of it is not a root of a polynomial dividing f(y) of a lower degree. No matter what 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?