On the sum of two integral squares in quadratic fields Q ( √ ± p )
On the sum of two integral squares in quadratic fields Q ( √ ± p )
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发表时间：
2011
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通讯作者：
Dasheng Wei;D. Wei
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作者：
Dasheng Wei;D. Wei
他的方法本质上依赖于这样一个事实：当d是上述整数之一时，域Q（√d,√- d）的类数为1。然而，这种方法并不适用于一般的二次域。最近，Harari[1]证明了在一般纤维为环面的主齐次空间的数域的整数环上，Brauer-Manin阻塞是阻碍整点存在的唯一障碍。然而，[1]给出的Brauer - Manin阻碍不是建设性的，不能用该结果来确定该方案的积分点是否存在。费旭和作者在[6]和[7]中给出了另一个建设性的证明。本文将[6]方法应用于具有p撇的二次域Q（√p）和Q（√−p）。符号和术语是标准的，如果没有解释。设F是一个数字域，F的整数环，ΩF是F中所有素数的集合，∞是F中所有无限素数的集合。为简单起见，我们将p∈ΩF \∞写成p <∞。设Fp为F在p处的补全，oFp为每个p∈ΩF of在p处的局部补全。对于p∈∞，写oFp = Fp。我们也表示阿黛尔戒指（见图）。F的理想环由AF（响应）。IF)，并设置
His method essentially depends on the fact that the class number of the field Q( √ d, √ −d) is 1 when d is one of the above integers. However, this method does not apply to general quadratic fields. Recently, Harari [1] showed that the Brauer–Manin obstruction is the only obstruction to existence of integral points of a scheme over the ring of integers of a number field whose generic fiber is a principal homogeneous space of a torus. However, the Brauer– Manin obstruction given in [1] is not constructive and one cannot use that result to determine the existence of integral points for the scheme. Fei Xu and the author gave another, constructive proof of that result in [6] and [7]. In this paper we apply the method of [6] to the quadratic fields Q(√p) and Q( √ −p) with p prime. The notation and terminology are standard if not explained. Let F be a number field, oF the ring of integers of F , ΩF the set of all primes in F , and ∞ the set of all infinite primes in F . For simplicity, we write p < ∞ for p ∈ ΩF \ ∞. Let Fp be the completion of F at p, and oFp be the local completion of oF at p for each p ∈ ΩF . Write oFp = Fp for p ∈ ∞. We also denote the adele ring (resp. the idele ring) of F by AF (resp. IF ), and set