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See DF Section 2.3.

Euclid’s Algorithm

  1. Well-ordering of the integers. Every nonempty set of positive integers has a least element.
  2. Let $a$ and $b$ be nonzero integers. Consider the set
\[X= \{ax+by: x,y\in\Z \}\]
  1. $X$ contains positive elements.
  2. $X$ contains a smallest positive element, call it $d$. So $ax+by=d$.
  3. Note that $X$ contains every (positive and negative) multiple of $d$.
  4. Take any other positive element $z$ of $X$. Then $z=qd+r$ but also $z=ax’+by’$. Suppose $r>0.$ So $qd+r=ax’+by’$ and therefore $r=a(x’-qx)+b(y’-qy)$. This means that $r$ is in $X$; but since $r$ is less than $d$, this cannot happen. It follows that $r=0$ and every element of $X$ is a multiple of $d$.
  5. We conclude that $X=d\Z$.
  6. Since $X$ contains both $a$ and $b$, we see that $d$ is a common divisor of $a$ and $b$.
  7. If $g$ is any other common divisor of $a$ and $b$, then $g$ divides $d$ since $d=ax+by$.
  8. Therefore $d$ is the greatest common divisor of $a$ and $b$, and any other common divisor of $a$ and $b$ is a divisor of $d$.
  9. $a$ and $b$ are called relatively prime if their greatest common divisor is $1$.

Theorem: Given nonzero integers $x$ and $y$, the equation $ax+by=z$ has a solution if and only if $z$ is a multiple of the greatest common divisor of $a$ and $b$.

Corollary 1: If $a$ and $n$ are relatively prime, and $a\mid nb$, then $a\mid b$.

Proof: Write $ax+ny=1.$ Multiply by $b$ to get $abx+nby=b$. Since $a$ divides both terms on the left, it divides $b$.

Corollary 2: Given integers $a$ and $b$ with greatest common divisor $d$, the integers $a/d$ and $b/d$ are relatively prime (i.e. have gcd equal to one).

Proof: Divide $ax+by=d$ by $d$.

Corollary 3: The least common multiple of $m$ and $n$ is $mn/d$ where $d=\gcd(m,n)$.

Proof: Suppose $x$ is a common multiple of $m$ and $n$. Write $x=am$. Then $n\mid x$ so $n\mid am$ and therefore $\frac{n}{d}\mid a\frac{m}{d}$. By Corollaries 1 and 2 this means $\frac{n}{d}$ divides $a$, so $mn/d$ divides $x$. Thus $mn/d$ is the least common multiple and any common multiple is a multiple of $mn/d$.

Congruences

Theorem: The congruence equation \(ax\equiv b\pmod{n}\) has solutions if and only if $d=\mathrm{gcd}(a,n)$ divides $b$. In that case it has $n/d$ solutions modulo $n$.

Proof: Solving the congruence equation \(ax\equiv b\pmod{n}\) is equivalent to solving the equation \(ax+ny= b.\)

Euclid’s algorithm tells us this equation has a solution if and only if $d=\gcd{a}{n}$ divides $b$. When this holds, swe have a solution to our congruence $x$ to our congruence. Notice that $x+k\frac{n}{d}$ is also a solution to this equation for $k=0,\ldots, d-1$, so we actually have $d$ solutions. If $x$ and $x’$ are any two solutions to this equation, then subtracting $ax+ny=b$ from $ax’+ny’=b$ yields \(a(x-x')+n(y-y')=0\) and so, since $n/d$ and $a/d$ have gcd equal to one we conclude that $x-x’$ is divisible by $n/d$. Therefore we have found all solutions.

Cyclic Groups

Key Facts about Cyclic Groups

  1. Any cyclic group of infinite order is isomorphic to $\Z$. Any cyclic group of finite order $n$ is isomoprhic to $\Zn{n}$. A group $G$ is cyclic if and only if there is a surjective homomorphism from $\Z$ to $G$.
  2. An infinite cyclic group has two generators.
  3. If $g$ is a generator of a finite cyclic group $G$ of order $n$, then $g^{a}$ has order $n/\mathrm{gcd}(n,a)$. Thus $g^{a}$ generates $G$ if and only if $\mathrm{gcd}(n,a)=1$.
  4. Every subgroup of $\Zn{n}$ is cyclic, and there is a unique such subgroup for every $d\mid n$.
  5. If $H$ is the cyclic subgroup of $\Zn{n}$ of order $d$ where $d \mid n$, then the elements of $H$ are the multiples of $n/d$. The generators of $H$ are the multiples $kn/d$ where $\mathrm{gcd}(k,d)=1$.
  6. If $\mathrm{gcd}(n,m)=1$ then $\Zn{nm}$ is isomorphic to $\Zn{n}\times\Zn{m}$.
  7. If $n$ and $m$ are relatively prime, then a pair $(a,b)$ generates $\Zn{n}\times\Zn{m}$ if and only if $a$ and $b$ generate $\Zn{n}$ and $\Zn{m}$ respectively. Therefore $\phi(nm)=\phi(n)\phi(m)$ when $n$ and $m$ are relatively prime.
  8. If $n=p_1^{e_1} p_2^{e_2}\cdots p_k^{e_k}$ where the $p_i$ are distinct primes then
\[\phi(n)=\prod_{i=1}^{k} (p^{e_i}-p^{e_{i}-1})=n\prod_{p|n}(1-\frac{1}{p})\]

For discussion

  1. We know that $\Zn{6}$ is a subgroup of $\Zn{24}$. Find all injective maps $\Zn{3}\to\Zn{12}$.
  2. “Reduction mod 6” gives a surjective homomorphism $\Zn{24}\to\Zn{6}$. Find the inverse image of $5$ under this map.
  3. Find all surjective homomorphisms $\Zn{24}\to\Zn{6}$.
  4. Prove that $(\Zn{11})^{\times}$ is cyclic. In fact $(\Zn{p})^{\times}$ is always cyclic, we’ll prove this later.

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