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\(A_4\)-sextic fields with a power basis. (English) Zbl 1229.11136

From the introduction: Let \(K\) be an algebraic number field of degree \(n\). Let \(O_K\) denote the ring of integers of \(K\). The field \(K\) is said to possess a power basis if there exists an element \(\theta\in O_K\) such that \(O_K = \mathbb Z+\mathbb Z\theta+\dots+\mathbb Z\theta^{n-1}\). A field having a power basis is called monogenic. For an extended history of monogenic number fields the reader should consult [I. Gaál, Diophantine equations and power integral bases. New computational methods, Boston: Birkhäuser (2002; Zbl 1016.11059)]. In this paper we exhibit infinitely many monogenic sextic fields with Galois group \(A_4\). We prove the following result.
Theorem. Let \(d\in \mathbb Z\). Set
\[ f_d(x):= x^6 + (2d + 2)x^4 + (2d-1)x^2-1\in \mathbb Z[x].\tag{1} \]
Let \(\theta_d\in \mathbb C\) be a root of \(f_d(x)\). Set \(K_d = \mathbb Q(\theta_d)\). Then \[ [K_d : \mathbb Q] = 6,\quad \text{Gal}(f_d)\simeq A_4, \] and the fields \(K_d\) \((d\in\mathbb Z)\) are distinct. Moreover \(K_d\) is monogenic with ring of integers \(\mathbb Z[\theta_d]\) for infinitely many values of \(d\).
We prove that \(K_d\) is monogenic whenever \(4d^2 + 2d + 7\) is squarefree, which occurs for infinitely many values of \(d\) by a result of T. Nagel [Hamb. Abh. 1, 179–194 (1922; JFM 48.0132.04)].
We remark that H. Anai and T. Kondo [RIMS Kokyuroku 941, 57–72 (1996; Zbl 1040.12500)] have stated without proof that
\[ \text{Gal}(x^6+(2a+2)x^4+(2a-1)x^2 -1) \simeq A_4 \] for all \(a\in\mathbb Q\) for which \(x^6 + (2a+2)x^4 + (2a-1)x^2 - 1\) is irreducible in \(\mathbb Q[x]\). We show in Section 2 that \(f_d(x)\) is, in fact, irreducible in \(\mathbb Z[x]\) for all \(d\in\mathbb Z\) (Lemma 2.2) and that \(\text{Gal}(f_d)\simeq A_4\) for all \(d\in\mathbb Z\) (Lemma 2.4). In Section 3 we complete the proof of the theorem.

MSC:

11R21 Other number fields
11R04 Algebraic numbers; rings of algebraic integers
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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Full Text: Euclid