## $$A_4$$-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

### Citations:

JFM 48.0132.04; Zbl 1016.11059; Zbl 1040.12500
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