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**Circular units of real abelian fields with four ramified primes.**
*(English)*
Zbl 1424.11155

The author studies the group of circular numbers and the group of circular units in Sinnott’s sense of real finite abelian extensions of \(\mathbb{Q}\) with exactly four ramified primes, with the aim to construct \(\mathbb{Z}\)-bases of these groups in five special infinite families of cases. To give an example of these cases let us describe the case which is proven in the paper (the proofs of all other four cases can be found in the thesis [Circular units of abelian fields. Master’s thesis. Brno: Masaryk University (2017)] of the author):

Let \(k/\mathbb{Q}\) be a real abelian extension having exactly four ramified primes \(p_1,p_2,p_3,p_4\). Let \(K\) be the genus field of \(k\) in the narrow sense and let \(K_i\) be the maximal subfield of \(K\) ramified only at \(p_i\) over \(\mathbb{Q}\). Let us assume that the extension \(K/k\) and also the extensions \(K_i/\mathbb{Q}\) for each \(i=1,2,3,4\) are cyclic. Let us denote \(r_i=[K:kK_i]\) and \(a_i=[(k\cap K_i):\mathbb{Q}]\). The case proven in the paper consists in the following assumptions: \(a_1=a_2=a_3=r_4=1\), \(r_1\ne1\), \(r_2\ne1\), \(r_3\ne1\), \([K:k]=r_1r_2r_3\) and the numbers \(r_1\), \(r_2\), \(r_3\) are pairwise coprime. Under these assumptions the author describes a \(\mathbb{Z}\)-basis of the group of circular numbers of \(k\) and a \(\mathbb{Z}\)-basis of the group of circular units of \(k\). He also studies of the module of all relations satisfied by the generators of the group of circular numbers of \(k\), and shows that the quotient of this module by the submodule generated by the norm relations is generated by Ennola relations coming from the subfields of \(k\) ramified over \(\mathbb{Q}\) at three primes and that the Galois action on this quotient is trivial.

Let \(k/\mathbb{Q}\) be a real abelian extension having exactly four ramified primes \(p_1,p_2,p_3,p_4\). Let \(K\) be the genus field of \(k\) in the narrow sense and let \(K_i\) be the maximal subfield of \(K\) ramified only at \(p_i\) over \(\mathbb{Q}\). Let us assume that the extension \(K/k\) and also the extensions \(K_i/\mathbb{Q}\) for each \(i=1,2,3,4\) are cyclic. Let us denote \(r_i=[K:kK_i]\) and \(a_i=[(k\cap K_i):\mathbb{Q}]\). The case proven in the paper consists in the following assumptions: \(a_1=a_2=a_3=r_4=1\), \(r_1\ne1\), \(r_2\ne1\), \(r_3\ne1\), \([K:k]=r_1r_2r_3\) and the numbers \(r_1\), \(r_2\), \(r_3\) are pairwise coprime. Under these assumptions the author describes a \(\mathbb{Z}\)-basis of the group of circular numbers of \(k\) and a \(\mathbb{Z}\)-basis of the group of circular units of \(k\). He also studies of the module of all relations satisfied by the generators of the group of circular numbers of \(k\), and shows that the quotient of this module by the submodule generated by the norm relations is generated by Ennola relations coming from the subfields of \(k\) ramified over \(\mathbb{Q}\) at three primes and that the Galois action on this quotient is trivial.

Reviewer: Radan Kučera (Brno)

### MSC:

11R20 | Other abelian and metabelian extensions |