Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 20 September 2014

Lecture 6: Greatest common divisors and Euclid's algorithm

Number systems – $𝔽\left[t\right]$ polynomials

Let $𝔽$ be a field. $$𝔽\left[t\right]=\{a_0+a_{1}t+a_2{t}^2+\cdots \hspace{0.17em}|\hspace{0.17em}{a}_i\in 𝔽,\text{all but a finite number of}\hspace{0.17em}{a}_i\hspace{0.17em}\text{equal}\hspace{0.17em}\text{<mn>0</mn>}\}$$ with addition and product so that $$(5+t+7t^2)+(3+2t+8t^2+(-3)t^3)=8+3t+15t^2+4t^3,$$ $$\begin{array}{ccc}{\displaystyle (1+2t^2+t^3)(0+0t+t^2+2t^3)}& =& {\displaystyle t^2+2t^3+4t^4+8t^5+t^5+2t^6}\\ & =& {\displaystyle t^2+2t^3+4t^4+9t^5+2t^6,}\end{array}$$ for example.

Let $d\in 𝔽\left[t\right]\text{.}$ The ideal generated by $d$, or the set of multiples of $d$, is $$d𝔽\left[t\right]=\left\{dp\hspace{0.17em}\right|\hspace{0.17em}p\in 𝔽\left[t\right]\}\text{.}$$ For example, $t^2+2t^3+4t^4+9t^5+2t^6\in (1+2t^2+t^3)𝔽\left[t\right]\text{.}$

Let $a,d\in 𝔽\left[t\right]\text{.}$ The polynomial $d$ divides $a$, $d|a,$ if $a\in d\mathbb{Z}\text{.}$

$$(1+2t^2+t^3)|(t^2+2t^3+4t^4+9t^5+2t^6)\text{.}$$

Let $x,m\in 𝔽\left[t\right]\text{.}$ The greatest common divisor of $x$ and $m,$ $\text{gcd}(x,m)$ is a monic polynomial $d$ such that

(a)	$d|x$ and $d|m\text{,}$
(b)	If $\ell \in {\mathbb{Z}}_{>0}$ and $\ell |x$ and $\ell |m$ then $\ell |d\text{.}$

Let $p\in 𝔽\left[t\right],$ $p=p_0+p_{1}t+p_2{t}^2+\cdots$ with $p\ne 0\text{.}$ The degree of $p$ is $N\in {\mathbb{Z}}_{>0}$ such that $p_N\ne 0$ and if $k\in {\mathbb{Z}}_{>0}$ and $k>N$ then $p_k=0\text{.}$

A polynomial $p=p_0+p_{1}t+p_2{t}^2+\cdots$ is monic if $p_N=1,$ where $N=\text{deg}\left(p\right)\text{.}$

(Euclid's algorithm) Let $a,b\in 𝔽\left[t\right]\text{.}$ There exist $q,r\in 𝔽\left[t\right]$ such that

(a)	$a=bq+r,$
(b)	Either $r=0$ or $\text{deg}\left(r\right)<\text{deg}\left(b\right)\text{.}$

Let $x,m\in 𝔽\left[t\right]\text{.}$

(a)	There exists a monic polynomial $\ell$ such that $$\ell 𝔽\left[t\right]=x𝔽\left[t\right]+m𝔽\left[t\right]\text{.}$$
(b)	Let $d=\text{gcd}(x,m)\text{.}$ Then $$d=\ell \text{.}$$

Find $$\text{gcd}((x^4-3x^3+3x^2-3x+2),(x^3-10x^2+23x-14))\text{.}$$ $$\begin{array}{cc}& \phantom{\text{)}}{x}^{\phantom{4}}+7\\ x^3-10x^2+23x-14& \overline{\text{)}{x}^4-3x^3+3x^2-3x+2\phantom{+1000}}\\ & \phantom{\text{)}}{x}^4-10x^3+23x^2-14x\\ & \overline{\phantom{\text{)}{x}^4-}7x^3-20x^2+11x+2\phantom{+10}}\\ & \phantom{\text{)}{x}^4-}7x^3-70x^2+161x-98\\ & \phantom{\text{)}{x}^4-}\overline{\phantom{7x^3-}50x^2-150x+100}\end{array}$$ So $$\begin{array}{ccc}{\displaystyle (x^4+3x^3+3x^2-3x+2)}& =& {\displaystyle (x^3-10x^2+23x-14)(x+7)+50(x^2-3x+2),}\\ {\displaystyle x^3-10x^2+23x-14}& =& {\displaystyle (x^2-3x+2)(x-7)\text{.}}\end{array}$$ So $$\text{gcd}(x^4-3x^3+3x^2-3x+2,x^3-10x^2+23x-14)=x^2-3x+2$$ and $$x^2-3x+2=\frac{1}{50}(x^4-3x^3+3x^2-3x+2)+(\frac{-1}{50}x-\frac{7}{50})(x^3-10x^2+23x-14)\text{.}$$

Notes and References

These are a typed copy of Lecture 6 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on August 5, 2011.

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