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We now specialise the results of the previous section to the case of
extensions of
of degree 2. Such a field is of the form
[*T*]/*P*(*T*) where *P*(*T*) is an irreducible polynomial of degree 2.
An order in such a field is generated by 1 and a non-rational element
that satisfies an equation of the type
*P*(*T*) = *T*^{2} - *bT* + *c*. Thus
every order has the form *R*[*T*]/*P*(*T*). Now, it is clear that
Trace() = *b* and
Nm() = *c*. Moreover,
Trace() = Trace(*b* - *c*) = *b*^{2} - 2*c*. Thus the discriminant
*D*_{R} of *R* is the determinant of
which is *b*^{2} - 4*c* (as expected). In particular, we see that
*D*_{R} = *b*^{2}(mod 4); i. e. the discriminant must be 0 or 1 modulo 4. In
the first case, we can replace by
+ (*b* - 1)/2 so that we
get an element with trace 1. In the second case, we can replace
by
+ *b*/2 to get an element with trace 0. Thus we can
assume that the equation takes the form *T*^{2} - *T* + *N* in the first case
and *T*^{2} + *N* in the second case. An alternative normalisation is to
replace by
= (*D*_{R} + )/2 in both cases; this
can be done since *D*_{R} + *b* is even in both cases. We thus have a
natural basis for *R*. There is also a natural involution on *R* which
sends
to
- or equivalently to
*D*_{R} - .

**Subsections**

** Next:** 7.1 Prime ideals
** Up:** Some Lectures on Number
** Previous:** 6.7 Prime ideals
Kapil Hari Paranjape
2002-10-20