Indecomposable integers in real quadratic fields

Abstract

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q √ D where D > 1 is a squarefree integer. Their conjecture was later disproved by Kala for D ≡ 2 mod 4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D ≡ 1, 2, 3 mod 4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O √ D .

On Kitaoka's conjecture and lifting problem for universal quadratic forms

“…Indecomposable algebraic integers turned out to be one of the key tools in the recent advances on this topic. In this short paper, we first prove in Theorem 5 that each indecomposable has norm smaller or equal to the discriminant of $$F$$, significantly extending previous results in the quadratic and cubic cases [9, 15, 32], as well as improving the previous general bound [4], which is typically much worse than ours, as it depends on the regulator. Our result is also a substantial step toward answering [24, Problem 53].…”

“…This leads to the Pythagoras number, a constant well‐studied also in other settings (see, for example, [23]). Yet, in the case of the ring of integers of a totally real number field, all that is known in general is that the Pythagoras number is finite [28] and bounded by the degree of the field [16], but can grow arbitrarily large [28] (in the non‐totally real case, the Pythagoras number $$\leqslant 5$$ [27]; for some small degree cases see [22, 26, 31]). Furthermore, Siegel [29] proved that for each number field $$F$$ there exists $$m\in \mathbb {N}$$ such that all totally positive integers divisible by $$m$$ can be represented as the sum of squares.…”

There are no universal ternary quadratic forms over biquadratic fields

“…Despite this fact, there is not much known about them. In quadratic fields, they were characterized by Perron [22], Dress and Scharlau [8], and their norms were studied by several authors [11,14,27]. Some general statements can be found in the work of Brunotte [4].…”

There are no universal ternary quadratic forms over biquadratic fields

“…Despite this fact, there is not much known about them. In quadratic fields, they were characterized by Perron [Pe], Dress and Scharlau [DS], and their norms were studied by several authors [JK,Ka2,TV]. Some general statements can be found in the work of Brunotte [Bru].…”