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The class number one problem for some non-abelian normal CM-fields of degree 24

F. Lemmermeyer, S. Louboutin, R. Okazaki (1999)

Journal de théorie des nombres de Bordeaux

We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to 𝒜 4 , the alternating group of degree 4 and order 12 . There are two such fields with Galois group 𝒜 4 × 𝒞 2 (see Theorem 14) and at most one with Galois group SL 2 ( 𝔽 3 ) (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number 1 .

The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields

Takashi Fukuda, Hisao Taya (1995)

Acta Arithmetica

1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension k of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of k / k . We know by the Ferrero-Washington...

The Ljunggren equation revisited

Konstantinos A. Draziotis (2007)

Colloquium Mathematicae

We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.

The minimal resultant locus

Robert Rumely (2015)

Acta Arithmetica

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map γ o r d ( R e s ( φ γ ) ) factors through a function o r d R e s φ ( · ) on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in P ¹ K , or on a segment, and the minimal resultant locus is contained in the tree in P ¹ K spanned by the fixed points and poles...

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