minor updates

abstract-algebra-charles-pinter-notes.tex

@@ -285,7 +285,7 @@ Quotient group construction is useful as a way of actually manufacturing all the
 \section{Rings}
 
 \begin{definition}[Ring]
-    A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axions:
+    A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axioms:
     \begin{enumerate}[i.]
 	\item $A$ with addition alone is an abelian group.
 	\item Multiplication is associative.
@@ -348,7 +348,7 @@ Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}
     If there is no such positive integer $n$, $A$ has characteristic $0$.
 \end{definition}
 
-\section{Factoring into primes}
+\section{Elements of number theory}
 
 \begin{definition}[Euclid's lemma]
     Let $m$ and $n$ be integers, and let $p$ be a prime. If $p|(mn)$, then either $p|m$ or $p|n$.
@@ -366,9 +366,6 @@ Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}
 
 From the last two theorems: every integer $m$ can be factored into primes, and the prime factors of $m$ are unique (except for the order).
 
-
-\section{Elements of number theory}
-
 \begin{theorem}[Little theorem of Fermat]
     Let $p$ be a prime. Then,
     $$a^{p-1} \equiv 1 \pmod p, \forall a \not\equiv 0 \pmod p$$
@@ -381,7 +378,7 @@ From the last two theorems: every integer $m$ can be factored into primes, and t
 \end{theorem}
 
 \begin{theorem}[Euler's theorem]
-    If $a$ and $n$ are relatively prime, $a^{\phi(n)} \equiv 1 \pmod n$.
+    If $a$ and $n$ are relatively prime, $$a^{\phi(n)} \equiv 1 \pmod n$$
 \end{theorem}
 
 

number-theory.sage

@@ -1,8 +1,12 @@
 # Chinese Remainder Theorem
-def crt(a_i, m_i, M):
+def crt(a_i, m_i):
     if len(a_i)!=len(m_i):
         raise Exception("error, a_i and m_i must be of the same length")
 
+    M=1
+    for i in range(len(m_i)):
+        M = M * m_i[i]
+
     x = 0
     for i in range(len(a_i)):
         M_i = M/m_i[i]

number-theory_test.sage

@@ -4,13 +4,11 @@ load("number-theory.sage")
 # Chinese Remainder Theorem tests
 a_i = [5, 3, 10]
 m_i = [7, 11, 13]
-M = 1001
-assert crt(a_i, m_i, M) == 894
+assert crt(a_i, m_i) == 894
 
 a_i = [3, 8]
 m_i = [13, 17]
-M = 221
-assert crt(a_i, m_i, M) == 42
+assert crt(a_i, m_i) == 42
 
 
 #####