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