## Norm of an ideal

Dear Professor Kim,

In the document ‘Polynomial rings and their quotients’ on the Algebraic Number Theory course webpage, there is an example at the bottom of page 2 where the penultimate line reads

‘and $N(J)=1$‘.

Is it supposed to be N(J)=0 since we calculated that the quotient ring was isomorphic to 0?

I’m sorry if this is a minor point, but at this stage even the smallest things can confuse me!

Many Thanks

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Reply:

I thought I would post this to clear up a common logical confusion. I will just pose the question: given a number field $F$, its ring of integers $O_F$, and an ideal $J\subset O_F$, what is the definition of the norm of $J$?

### One Comment

1. Wai-Kae
Posted May 1, 2009 at 10:51 am | Permalink | Reply

I think I see it now,
The definition is

N(J) = |O_F / J |

So in the example,

Z[i]/(5,1+i) = {0}

=> N(J) = |{0}| = 1

I’m afraid I might have overseen this amidst the chaos… sorry..!..