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Let *R* denote the subring of complex numbers generated by
= exp(2/*p*); let *K* denote the quotient field of *R*,
which is called the cyclotomic field of *p*-th roots of unity. We
review some well-known facts about the ring *R* and the field
*K*--mostly without proof.
The ring *R* is isomorphic to
*Z*[*X*]/((*X*)), where
(

*X*) =

*X*^{p - 1} +

^{ ... } +

*X* + 1 = (

*X*^{p} - 1)/(

*X* - 1)

is an irreducible polynomial (a simple application of the Eisenstein
criterion). The field *K* is a Galois extension of *Q* with Galois
group
*F*_{p}^{*}; this is a cyclic group of order (*p* - 1). We use
for a fixed choice of generator. We use
to denote
() since
is the
restriction of complex conjugation to *R*.
The ring *R* is a Dedekind domain, i. e. unique factorization holds for
ideals. The prime ideals in this ring are described as follows:
- If
*q* *Z* is a prime number different from *p*. Then let
*f* be the order of *q* in
*F*_{p}^{*} and let *g* = (*p* - 1)/*f*. Then
there are *g* prime ideals
*Q*_{1},..., *Q*_{g} in *R* such that their
norms are *q*^{f}.
- The element
= 1 - is prime in
*R* and
= (unit)^{ . }*p*.

A closed form expression for the generators of the group *U* of units
of *R* is not known. However, the numbers
*u*_{j} =

(

)/

= 1 +

+

^{ ... } +

are in *R* and are units there. The subgroup
*U*_{cycl} of the
group *U* of units of *R* generated by the *u*_{j} for
*j* = 2,...,(*p* - 1)
is called the group of cyclotomic units. If *u* is a unit in *R*,
then
/*u* is a root of unity in *R*. The roots of unity in
*R* are all of the form
± for some
*j* = 0,..., *p* - 1. An
element of *R* is a *p*-th power only if it is congruent to an integer
modulo *pR*. It follows that
/*u* = for some *j*
(i. e. there is no minus sign).
Let *L* denote the subfield of *K* fixed by complex conjugation; let
*S* = *L* *R*. Then *L* is a Galois extension of *Q* with Galois group
*F*_{p}^{*}/{±1}. No complex embeddings of *K* have image within
real numbers while all complex embeddings of *L* have image within
real numbers; in other words, *K* is purely imaginary and *L* is
totally real. Again, *S* is a Dedekind domain and its ideals are
described as follows:
- If
*q* *Z* is a prime number different from *p*. Then let
*f'* be the order of *q* in
*F*_{p}^{*}/{±1} and let
*g'* = (*p* - 1)/2*f'*. Then there are *g'* prime ideals
*Q*_{1},..., *Q*_{g} in
*R* such that their norms are *q*^{f'}.
- The element
= 1 - ( + ) is prime in
*R* and
= (unit)^{ . }*p*.

If *u* is a unit in *R*, then we have seen that
/*u* = for some integer *r*. But then
*r* 2*s*(mod *p*) for some integer *s*; hence
*u*_{1} = *u* is in
*S*. Hence, any unit in *R* is the product of a root of unity and a
unit in *S*.
If *I* is any ideal in *S* then *IR* is principal in *R* if and only
if *I* is principal in *S*. Hence the homomorphism from the class
group of *S* to that of *R* is injective. In particular the order *h*
of the class group of *R* is divisible by the order *h*_{+} of the
class group of *S*.
If we have a unit *u* in *R* such that it is congruent to an integer
modulo *pR* and if *u* is itself *not* a *p*-th power, then the
field extension of *K* obtained by adjoining a *p*-th root of *u* is a
cyclic extension of *K* of order *p* which is unramified everywhere.
Finally we have a fact from Class Field theory. If there is an ideal
*I* in *R* such that *I*^{p} is principal and *I* is not principal, then
there is a cyclic entension of *K* of order *p* which is unramified
everywhere. This follows from the identification of the class group of
*R* with the Galois group of the maximal unramfied abelian extension
of *K*. Now we use the fact that if an abelian group has an element of
order *p*, then it has a non-trivial character of order *p*.

** Next:** 2 Construction of cyclic
** Up:** Kummer's proof of Fermat's
** Previous:** Fermat's Last Theorem:
Kapil Hari Paranjape
2002-11-22