曲線的函數體的位階與畢達哥拉斯數

Abstract

在本篇文章的第二章中，我們收集了Witt的文章當中的一些結論，在其中證明了曲線在R 上的函數體的畢達哥拉斯數小於等於2。接著在第三章中我們考慮一般體上的函數體，計算圓錐曲線x^2+y^2+n在體K上的函數體F_{n/K}的位階，並證明了若 s(F_{n/K})≦2，則 p(F_{n/k})=2 當且僅當s(K)=1或者K有遺傳歐幾里得性質。此外，證明了當K=Q時，若n≡3,6,7(mod 8) 為無平方因子的正整數，則p(F_{n/Q})=5。在第三章的最後，我們證明了對於m=1，以下敘述成立：x^{2k}+n在Q(x)中是m+1個平方和當且僅當n在Q中是m個平方和；然而此敘述對於m=2,3不成立。第四章中收集了Pourchet的結論，之中證明了對於代數數體K，K(x)的畢達哥拉斯數小於等於5且Q(x)的畢達哥拉斯數恰為5。最後在第五章中我們證明了對於m=1,2，以下敘述成立：對於整係數多項式f(x)，f在Q(x)中是m 個平方和當且僅當對於所有正整數n都有f(n)在Q中是m個平方和；然而此敘述對於m=3不成立。

In this paper, we collect some results of Witt’s paper in Chapter 2, which states that the pythagoras numbers of function fields of curves over R are less than 2. Next we consider the function field over general fields in Chapter 3, we compute the levels and pythagoras numbers of the function fields F_{n/K} of the conics x^2+y^2+n over a field K, and prove that if s(F_{n/K})≦2, then p(F_{n/K})=2 if and only if s(K)=1 or K is hereditarily euclidean. Moreover, for K = Q, we show that p(F_{n/Q})=5 for squarefree positive integer n with n≡3,6,7(mod 8). At the end of Chapter 3, we prove that x^{2k}+n is the sum of m+1 squares in Q(x) if and only if n is the sum of m squares for m=1 but false for m=2,3. In Chapter 4 we collect some results of Pourchet’s paper, which states that the pythagoras number of K(x) is less than 5 for number field K and it is exactly 5 if K=Q. Finally in Chapter 5 we show that for an integer-coefficient polynomial f, f is the sum of m squares in Q(x) if and only if f(n) is the sum of m squares for all positive integers n for m = 1, 2 but false for m = 3.