Diophantine Equations and Power Integral Bases

“…Ash, Brakenhoff, and Zarrabi give computational evidence supporting a conjecture of Lenstra [2] that suggests that monogenic fields are abundant. However, outside of the towers mentioned previously, the majority of results are known for extensions of small degree (see Gras [8], Nakahara [18], Shah [21], Gaál [6], among others). A general survey of recent results can be found in Narkiewicz [19, pp.…”

Section: Theorem 12 Let Be a Prime And Letmentioning

confidence: 99%

Discriminants of Chebyshev radical extensions

Gassert¹

2014

J. Théor. Nombres Bordeaux

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Let t be any integer and fix an odd prime ℓ. Let Φ(x) = T n ℓ (x) − t denote the n-fold composition of the Chebyshev polynomial of degree ℓ shifted by t. If this polynomial is irreducible, let K = Q(θ), where θ is a root of Φ. A theorem of Dedekind's gives a condition on t for which K is monogenic. For other values of t, we apply the Montes algorithm to obtain a formula for the discriminant of K and to compute basis elements for the ring of integers OK .

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“…in units x 1 , x 2 of K. This equation is called unit equation and plays an essential role in the resolution of several important types of diophantine equations, like Thue equations, norm form equations, index form equations etc, see [2].…”

Section: Introductionmentioning

confidence: 99%

“…M. M. de Weger [8] considered an analogous equation which can be treated as a generalization of unit equations. Let S be a finite set of primes and denote by H S the set of those integers which are only divisible by primes from S. Consider the equation 2 ∈ H S and y ∈ Z. The more general equation…”

Section: Introductionmentioning

confidence: 99%