Norms

Just a quick question.

The norm of an ideal, e.g. , is taken to be |N(a)|. I am not sure I understand what this means exactly. For instance, the norm of is 2, but i am not sure how they used this fact.
Thanks.

Also, when are u available for consultation from here onwards and is there a revision class planned? ————————————————————————————————————————
Hi again.
I sent you a question earlier regarding the norm of an ideal, but i think i may have confused myself (and possibly you!)

Firstly, I understand the use of N()=|N(a)| but i think, now, that we are to use a different equation for multiple element ideals. I think that is why I was confused for .
Also, i should have mentioned in the example that we were in Z[sqrt(-17)], but i think you figured that already.
————————————————————————————————————————

Reply:

Actually, the ideal wasn’t transmitted properly, but I can guess what it was.

First of all, the norm of an ideal is related to the norm of a number, but not the same:

If A is a non-zero ideal inside the ring R of algebraic integers in an algebraic number field K, then

N(A)=|R/A|

the number of elements inside the finite ring R/A. In particular, it is always a positive number. The relation with the norm of a number is that if A=(a), i.e., it is generated by a single element a, then

N(A)=|N(a)|

Note that the absolute value symbol here is used in the usual sense, and gives a positive number. When we used it above, it meant the cardinality of a set. You should get used to both notations and figure out what makes sense from the context.

When the ideal has many generators, the relation is more complicated. So for A=(a,b) we have

(a)\subset A,

and hence,

N(A) \leq N((a))=|N(a)|

or, more precisely,

N(A) divides |N(a)| using some theorem from the notes. (Find it!)

A basic case for computing norms is for a maximal ideal in the ring of integers of a field extension as described by Dedekind’s theorem. For the case

Q(\sqrt{-17})

you mention, the ring of integers is

Z[\sqrt{-17}],

so that we can compute the decomposition of ideals by consider the reductions of x^2+17. For example,

x^2+17 \equiv  (x-1)^2\ \  mod\  2,

so that

(2)=P_2^2

in Z[\sqrt{-17}]

for the maximal ideal P_2=(2, \sqrt{-17}-1). You’ll find in the discussion of Dedekind’s theorem a proof of the fact that N(P_2)=2.

The point that is

Z[\sqrt{-17}]/P_2

is a field extension of Z/2 of degree equal to the degree of x-1, i.e., 1. That is to say,

Z[\sqrt{-17}]/P_2 =Z/2

So

N(P_2)=|Z/2|=2.

Do a bunch of exercises that compute norms of numbers and ideals. A basic useful fact of course is that

N(IJ)=N(I)N(J)

for any (non-zero) $I,J$. Thus, once you have the norms of prime ideals, you have the norm of any ideal (provided you have the prime decomposition, of course). Of course, this fact is more commonly used the other way around, that is, to *compute* the prime decomposition.

What are the norms of the numbers \sqrt{2} and \sqrt{3} in

Q(\sqrt{2},\sqrt{3})?

What is the norm of the ideal

(2^{1/3}, 5)

in Z[2^{1/3}]?

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