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arnaucube 2 years ago
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a431ca9b92
4 changed files with 10 additions and 11 deletions
  1. BIN
      abstract-algebra-charles-pinter-notes.pdf
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      abstract-algebra-charles-pinter-notes.tex
  3. +5
    -1
      number-theory.sage
  4. +2
    -4
      number-theory_test.sage

BIN
abstract-algebra-charles-pinter-notes.pdf


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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}

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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]

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- 4
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
#####

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