Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updated: 20 September 2014

Lecture 6: Greatest common divisors and Euclid's algorithm

Number systems – 𝔽[t] polynomials

Let 𝔽 be a field. 𝔽[t]= { a0+a1t+ a2t2+ |ai 𝔽,all but a finite number ofai equal0 } with addition and product so that (5+t+7t2)+ (3+2t+8t2+(-3)t3) =8+3t+15t2+4t3, (1+2t2+t3) (0+0t+t2+2t3) = t2+2t3+4t4+ 8t5+t5+2t6 = t2+2t3+4t4+ 9t5+2t6, for example.

Let d𝔽[t]. The ideal generated by d, or the set of multiples of d, is d𝔽[t]= {dp|p𝔽[t]}. For example, t2+2t3+4t4+9t5+2t6(1+2t2+t3)𝔽[t].

Let a,d𝔽[t]. The polynomial d divides a, d|a, if ad.

(1+2t2+t3)| (t2+2t3+4t4+9t5+2t6).

Let x,m𝔽[t]. The greatest common divisor of x and m, gcd(x,m) is a monic polynomial d such that

(a) d|x and d|m,
(b) If >0 and |x and |m then |d.

Let p𝔽[t], p=p0+p1t+p2t2+ with p0. The degree of p is N>0 such that pN0 and if k>0 and k>N then pk=0.

A polynomial p=p0+p1t+p2t2+ is monic if pN=1, where N=deg(p).

(Euclid's algorithm) Let a,b𝔽[t]. There exist q,r𝔽[t] such that

(a) a=bq+r,
(b) Either r=0 or deg(r)<deg(b).

Let x,m𝔽[t].

(a) There exists a monic polynomial such that 𝔽[t]=x 𝔽[t]+m𝔽[t].
(b) Let d=gcd(x,m). Then d=.

Find gcd ( (x4-3x3+3x2-3x+2), (x3-10x2+23x-14) ) . )x4+7 x3-10x2+23x-14 )x4-3x3+3x2-3x+2+1000 ) x4-10x3+ 23x2-14x )x4- 7x3-20x2+ 11x+2+10 )x4- 7x3-70x2+161x-98 )x4- 7x3-50x2-150x+100 So (x4+3x3+3x2-3x+2) = (x3-10x2+23x-14) (x+7)+50(x2-3x+2), x3-10x2+23x-14 = (x2-3x+2)(x-7). So gcd(x4-3x3+3x2-3x+2,x3-10x2+23x-14) =x2-3x+2 and x2-3x+2=150 (x4-3x3+3x2-3x+2)+ (-150x-750) (x3-10x2+23x-14).

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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