2007-7.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	7	IB 2007 Groups, Rings and Modules	1.II.10G	2007

(i) State a structure theorem for finitely generated abelian groups.

(ii) If $K$ is a field and $f$ a polynomial of degree $n$ in one variable over $K$, what is the maximal number of zeroes of $f$ ? Justify your answer in terms of unique factorization in some polynomial ring, or otherwise.

(iii) Show that any finite subgroup of the multiplicative group of non-zero elements of a field is cyclic. Is this true if the subgroup is allowed to be infinite?