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Hyperplanes Without Squares in Cubic Fields
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DOI：10.1007/s40840-026-02089-3.png)
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Abstract
En 中文
In this paper, we show that for each cubic algebraic number alpha there is a nonzero gamma is an element of Q(alpha) such that for all u, v is an element of Q the product (u + v alpha)gamma is not a square in the field Q(alpha) except for the trivial case u = v = 0. This completes some earlier investigations in which such a result has been proved for most, but not all, cubic numbers alpha. The method is constructive and allows us to find such a number gamma explicitly. Our result implies that each cubic field K contains linear hyperplanes H that do not contain elements of the form w(2), where w is an element of K \ {0}. We show that the same is true for all totally real fields K as well. However, for each d >= 5 there are fields K of degree d such that each linear hyperplane H in K has an infinite intersection with K-2 = {w(2) w is an element of K}.
Keywords:
Cubic field
square in a field
irreducible polynomials
Hilbert's irreducibility theorem
Frobenius density theorem
Journal
B
IF:
1.2
Papers:
140
Citations:
0