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Remarks on the first case of Fermat’s theorem over number fields. (Remarques sur le premier cas du théorème de Fermat sur les corps de nombres.) (French. English summary) Zbl 1364.11082

Let \(K\) be a number field and \(O_K\) its ring of integers. Let \(p\) be an odd prime number. We say that the first case of Fermat’s theorem holds over \(K\) for the exponent \(p\), if there are not \( x,y,z\in O_K\) satisfying \(x^p + y^p + z^p = 0\) and \((xyz)O_K + pO_K = O_K\). In this paper, the first case of Fermat’s theorem is studied.
Suppose now that \(K\) is an imaginary quadratic field. For every integer \( n \geq 1\), we denote by \( W_n\) the resultant of polynomials \(X^n-1\) and \( (X + 1)^n-1\). Further, we denote by \(h_K\) the class number of \(K\). The author proves that the first case of Fermat’s theorem holds over \(K\) for the exponent \(p\), provided that \(p \not | h_K\) and there is an integer \( n \geq 1\) such that \(q = n p + 1\) is a prime which is decomposed in \( K\) and \(q \not |(n^n-1)W_n\). An interesting corollary is that the result is valid in case where \(p \not | h_K\) and \(2p + 1\) is a prime number which is decomposed in \(K\). Furthermore, some results for the field \(Q(i)\) are obtained.
Next, the author considers an arbitrary number field and proves that the first case of Fermat’s theorem holds over \(K\) for the exponent \(p\) is valid provided that there is a prime ideal of \(O_K\) above \(p\) having residual degree 1 and ramification index \(\leq p-1\), and for \( a = 1, \ldots, (p-3)/2\) we have \(1 + a^p \not \equiv (1 + a)^p \;\bmod p^2\). Furthermore, some interesting consequences of this result are given.

MSC:

11D41 Higher degree equations; Fermat’s equation
11R16 Cubic and quartic extensions
11R21 Other number fields

Citations:

Zbl 0538.10015
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