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126 lines
3.9 KiB
Markdown
126 lines
3.9 KiB
Markdown
# Math4302 Modern Algebra (Lecture 27)
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## Rings
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### Fermat’s and Euler’s Theorems
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Recall from last lecture, $ax\equiv b \mod n$, if $x\equiv y\mod n$, then $x$ is a solution if and only if $y$ is a solution.
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#### Theorem for existence of solution of modular equations
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$ax\equiv b\mod n$ has a solution if and only if $d=\operatorname{gcd}(a,n)|b$ And if there is a solution, then there are exactly $d$ solutions in $\mathbb{Z}_n$.
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<details>
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<summary>Proof</summary>
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For the forward direction, we proved if $ax\equiv b\mod n$ then $ax-b=ny$, $y\in\mathbb{Z}$.
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then $b=ax-ny$, $d|(ax-ny)$ implies that $d|b$.
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---
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For the backward direction, assume $d=\operatorname{gcd}(a,n)=1$. Then we need to show, there is exactly $1$ solution between $0$ and $n-1$.
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If $ax\equiv b\mod n$, then in $\mathbb{Z}_n$, $[a][x]=[b]$. (where $[a]$ denotes the remainder of $a$ by $n$ and $[b]$ denotes the remainder of $b$ by $n$)
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Since $\operatorname{gcd}(a,n)=1$, then $[a]$ is a unit in $\mathbb{Z}_n$, so we can multiply the above equation by the inverse of $[a]$. and get $[x]=[a]^{-1}[b]$.
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Now assume $d=\operatorname{gcd}(a,n)$ where $n$ is arbitrary. Then $a=a'd$, then $n=n'd$, with $\operatorname{gcd}(a',n')=1$.
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Also $d|b$ so $b=b'd$. So
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$$
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\begin{aligned}
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ax\equiv b \mod n&\iff n|(ax-b)\\
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&\iff n'd|(a'dx-b'd)\\
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&\iff n'|(a'x-b')\\
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&\iff a'x\equiv b'\mod n'
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\end{aligned}
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$$.
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Since $\operatorname{gcd}(a',n')=1$, there is a unique solution $x_0\in \mathbb{Z}_{n'}$. $0\leq x_0\leq n'+1$. Other solution in $\mathbb{Z}$ are of the form $x_0+kn'$ for $k\in \mathbb{Z}$.
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And there will be $d$ solutions in $\mathbb{Z}_n$,
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</details>
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<details>
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<summary>Examples</summary>
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Solve $12x\equiv 25\mod 7$.
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$12\equiv 5\mod 7$, $25\equiv 4\mod 7$. So the equation becomes $5x\equiv 4\mod 7$.
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$[5]^{-1}=3\in \mathbb{Z}_7$, so $[5][x]\equiv [4]$ implies $[x]\equiv [3][4]\equiv [5]\mod 7$.
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So solution in $\mathbb{Z}$ is $\{5+7k:k\in \mathbb{Z}\}$.
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---
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Solve $6x\equiv 32\mod 20$.
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$\operatorname{gcd}(6,20)=2$, so $6x\equiv 12\mod 20$ if and only if $3x\equiv 6\mod 10$.
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$[3]^{-1}=[7]\in \mathbb{Z}_{10}$, so $[3][x]\equiv [6]$ implies $[x]\equiv [7][6]\equiv [2]\mod 10$.
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So solution in $\mathbb{Z}_{20}$ is $[2]$ and $[12]$
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So solution in $\mathbb{Z}$ is $\{2+10k:k\in \mathbb{Z}\}$
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</details>
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### Ring homomorphisms
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#### Definition of ring homomorphism
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Let $R,S$ be two rings, $f:R\to S$ is a ring homomorphism if $\forall a,b\in R$,
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- $f(a+b)=f(a)+f(b)\implies f(0)=0, f(-a)=-f(a)$
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- $f(ab)=f(a)f(b)$
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#### Definition of ring isomorphism
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If $f$ is a ring homomorphism and a bijection, then $f$ is called a ring isomorphism.
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<details>
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<summary>Example</summary>
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Let $f:(\mathbb{Z},+,\times)\to(2\mathbb{Z},+,\times)$ by $f(a)=2a$.
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Is not a ring homomorphism since $f(ab)\neq f(a)f(b)$ in general.
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---
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Let $f:(\mathbb{Z},+,\times)\to(\mathbb{Z}_n,+,\times)$ by $f(a)=a\mod n$
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Is a ring homomorphism.
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</details>
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### Integral domains and their file fo fractions.
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Let $R$ be an integral domain: (i.e. $R$ is commutative with unity and no zero divisors).
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#### Definition of field of fractions
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If $R$ is an integral domain, we can construct a field containing $R$ called the field of fractions (or called field of quotients) of $R$.
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$$
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S=\{(a,b)|a,b\in R, b\neq 0\}
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$$
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a relation on $S$ is defined as follows:
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$(a,b)\sim (c,d)$ if and only if $ad=bc$.
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<details>
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<summary>This equivalence relation is well defined</summary>
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- Reflectivity: $(a,b)\sim (a,b)$ $ab=ab$
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- Symmetry: $(a,b)\sim (c,d)\Rightarrow (c,d)\sim (a,b)$
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- Transitivity: $(a,b)\sim (c,d)$ and $(c,d)\sim (e,f)\Rightarrow (a,b)\sim (e,f)$
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- $ad=bc$, and $cf=ed$, we want to conclude that $af=be$. since $ad=bc$, then $adf=bcf$, since $cf=ed$, then $cfb=edb$, therefore $adf=edb$.
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- Then $d(af-be)=0$ since $d\neq 0$ then $af=be$.
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</details>
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Then $S/\sim$ is a field. |