Dear Professor Kim,

I just have a few small questions. In lectures, we calculated the class group of , which has ring of algebraic integers . We then found that that (maximal ideal) ,

After a bunch of calculations we had to see whether was principal or not. Using the result,

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(non-zero) is principal iff there exists s.t

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We had to consider if there was s.t

Since the general element of the ideal is , is it ‘more correct’ to consider if there was s.t

I know this doesn’t make a whole lot of difference, it’s just one of those things.

So

which was equivalent to the statement

or

In general, do we consider any ‘modulo n’ so that the statement is simplified?

Many Thanks

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Reply:

You are absolutely right about the principality question. That is, when we have an ideal in the ring of integers of an algebraic number field such that , then we are led to consider solutions to the equation

for various . As we’ve seen in many examples, once is expressed in terms of an integral basis, this becomes an equation in variables with integer coefficients to which we can consider solutions. Now, if there are no solutions with , then, a fortiori, there are no solutions and we can conclude that is not principal. However, although I’m too lazy now to cook up an example, there are situations where there *is* a solution , but no solution . In the example you mention, the equation corresponding to is easily found to be

which is obviously more resrictive than

In our case, the latter already has no solution, so we don’t need to consider the more refined equation.

Regarding your second question, firstly, the equation is not *equivalent* to the congruence equation but just implies it. So if the congruence equation has no solution, neither does the original, which is how we used it. Now, I don’t quite understand your final question, but perhaps I should remark that considering congruences is a standard way of investigating solutions to quadratic equations. In fact, it is useful for *any* Diophantine equation. However, a rather deep theorem says that for quadratic equations, sufficiently many congruence equations completely determine whether or not the original equation has rational solutions.