1 Arithmetic of prime cyclotomic fields

Let R denote the subring of complex numbers generated by $\omega$ = exp(2$\pi$$\iota$/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 $\Bbb$Z[X]/($\Phi_{p}^{}$(X)), where

$$\Phi_{p}^{}$$(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 $\Bbb$Q with Galois group $\Bbb$F_p^*; this is a cyclic group of order (p - 1). We use $\gamma$ for a fixed choice of generator. We use $\overline{\alpha}$ to denote $\gamma^{(p-1)/2}_{}$($\alpha$) since $\gamma^{(p-1)/2}_{}$ 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:

1. If q $\in$ $\Bbb$Z is a prime number different from p. Then let f be the order of q in $\Bbb$F_p^* and let g = (p - 1)/f. Then there are g prime ideals Q₁,..., Q_g in R such that their norms are q^f.
2. The element $\lambda$ = 1 - $\omega$ is prime in R and $\lambda^{p-1}_{}$ = (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 = $$\gamma^{j}_{}$$($$\lambda$$)/$$\lambda$$ = 1 + $$\omega$$ + ^... + $$\omega^{j-1}_{}$$

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 $\overline{u}$/u is a root of unity in R. The roots of unity in R are all of the form ±$\omega^{j}_{}$ 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 $\overline{u}$/u = $\omega^{j}_{}$ for some j (i. e. there is no minus sign). Let L denote the subfield of K fixed by complex conjugation; let S = L $\cap$ R. Then L is a Galois extension of $\Bbb$Q with Galois group $\Bbb$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:

1. If q $\in$ $\Bbb$Z is a prime number different from p. Then let f' be the order of q in $\Bbb$F_p^*/{±1} and let g' = (p - 1)/2f'. Then there are g' prime ideals Q₁,..., Q_g in R such that their norms are q^f'.
2. The element $\mu$ = 1 - ($\omega$ + $\omega^{-1}_{}$) is prime in R and $\mu^{(p-1)/2}_{}$ = (unit) ^. p.

If u is a unit in R, then we have seen that $\overline{u}$/u = $\omega^{r}_{}$ for some integer r. But then r $\cong$ 2s(mod p) for some integer s; hence u₁ = $\omega^{-s}_{}$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.