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--- 2020-2021:teams:legal_string:数论概论学习小结_lgwza [2020/07/03 22:23]
lgwza [第 20 章欧拉\phi 函数与因数和]
+++ 2020-2021:teams:legal_string:数论概论学习小结_lgwza [2020/07/03 22:25] (当前版本)
lgwza [第 25 章二次互反律]
@@ 行 376: / 行 376: @@
 === 定理 25.1 (二次互反律) ===
-设 $p$, $q$ 是不同的奇素数, 则 $$ ()={
+设 $p$, $q$ 是不同的奇素数, 则
+$$
+\left(\frac{-1}{p}\right)=\left\{
+\begin{array}{rl}
+& if & p \equiv 1\pmod{4},\\
+-1 & if & p \equiv 3\pmod{4}.
+\end{array} \right.\\
-.\
+\left(\frac{2}{p}\right)=\left\{
+\begin{array}{rl}
+& if & p\equiv 1 \ or\ 7\pmod{8}, \\
+-1 & if & p\equiv 3\ or\ 5\pmod{8}.
+\end{array}\right.\\
-()={
+\left(\frac{q}{p}\right)=\left\{
+\begin{array}{rl}
-.\
+\left(\frac{p}{q}\right) & if &p\equiv 1\pmod{4}&or&q\equiv1\pmod{4},\\
+-\left(\frac{p}{q}\right)&if&p\equiv3\pmod{4}&and&q\equiv3\pmod{4}.
-()={
+\end{array}\right.
+$$
-. $$
 === 定理 25.2 (广义二次互反律) ===
-设 $a$, $b$ 为正奇数, 则 $$ ()={
+设 $a$, $b$ 为正奇数, 则
+$$
-.\
+\left(\frac{-1}{b}\right)=\left\{
+\begin{array}{rl}
-()={
+& if & b \equiv 1\pmod{4},\\
+-1 & if & b \equiv 3\pmod{4}.
-.\
+\end{array} \right.\\
-()={
+\left(\frac{2}{b}\right)=\left\{
+\begin{array}{rl}
+& if & b\equiv 1 \ or\ 7\pmod{8}, \\
+-1 & if & b\equiv 3\ or\ 5\pmod{8}.
+\end{array}\right.\\
-. $$
+\left(\frac{a}{b}\right)=\left\{
+\begin{array}{rl}
+\left(\frac{b}{a}\right) & if &a\equiv 1\pmod{4}\ or\ b\equiv1\pmod{4},\\
+-\left(\frac{b}{a}\right)&if&a\equiv b\equiv 3\pmod{4}.
+\end{array}\right.
+$$

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