Algebraic number theory, Marcus, Chapter 3, Question 9

Question

Question 9 in Marcus book.

Let $K$ and $L$ be the number field such that $K\subset L$ and let $R,S$ be their algebraic integers, respectively.

a) Let $I$ and $J$ be ideals in $R$, and suppose $IS|JS$. Showw that $I|J$.

(Suggestion(from the book): Factor $I$ and $J$ in primes in $R$)

b) Show that for each ideal $I$ in $R$, we $I=IS\cap R$.

(Set $J=IS\cap R$ and use a)

c)characterise those ideals $I$ of $S$ such that $I=(I\cap R)S$.

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Now, I need to understand something , if $P$ is any prime ideal in $R$, then what I can say about $PS$? I meant if $P$ is any prime ideal in the factorisation of $I$ , then is $PS\subset I$???

How to show $J=IS\cap R$?

Is there any website or book explain these concepts in more details and examples.

I really need to understand the concepts of this question.

Answer 1

It's not terrible, just prove ($2$) and then note $IS|JS \iff JS\subseteq IS$, now intersect both sides with $R$ and you get $J= R\cap SJ\subseteq R\cap IS = I$.

For ($2$) you get the intersection properties come to you since you can do it for prime powers, and then using the fact that distinct prime powers' intersections are their product.

For the last one, it's easy to see based on how primes split in the extension, so if $I=\prod_{i=1}^r\mathfrak{P}_i^{e_i}$ where if $\{\mathfrak{p}_j\}=\left\{\mathfrak{P}_i\cap R\right\}_{i=1}^r$ has that

$$\mathfrak{p}_jS=\left(\prod_{k=1}^{e_j}\mathfrak{P}_{i_k}^{e'_{i_k}}\right)^n$$

for every $s$ where $ne'_{i_k}=e_{i_k}$ for all $k$.