Real quadratic field

Consider the monogenic Real quadratic field where . alg

Sage

K.<α> = QuadraticField(223)

Discriminant

By Discriminant of an algebraic integer, we have

Group of units

Take the reduced element with simple continued fraction

whence

is the fundamental unit, and we have

Class group

Minkowski’s bound is given by

so applying Kummer’s factorization theorem:

			norms

Some algebraic integers of small field norm are

so

• from , we see ;
• from , we see ;
• from , wee see .

Therefore the ideal class group is cyclic of order 1 or 3. We show cannot be principal, whence .

Suppose towards contradiction for some , so . Then , so for some . It follows

Thus for , where by direct calculation

Now suppose , where , so both . We have

where the absolute value of the coëfficient of must be at least , leaving only the case . Thus , a prime number. Thus , which must be positive, must equal 1, so for some , which is impossible.

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