I am Arthur from Belgium student in 2nd year of mathematics and I am repeating the exercises for Algebra I, but this one extra exercise I just can't solve:
14.$\text{}$ i) If $R$ is a PID, $M=Rx_{1}+...+Rx_{N}$ a finitely generated $R$ module, $L\subset M$ a submodule of $R$. Then $L=Ry_{1}+...+Ry_{J}$, where $J\le N$ is also finitely generated.
ii) If $\mathbf{F}$ is a finite field with $q$ elements, and $N_{d} = N_{d}(\mathbf{F})$ is the number of normed polynomials $P \in \mathbf{F}[t]$ with degree $d$. What are $N_{2}$, $N_{3}$ ?
If there is anybody who understands these exercises, I would be very glad if you could explain them to me. Because when my professor tried it, I don't seem to have understood. Thank you for your attention.