Let F be a field. Recall that if f(T) and g(T) are polynomials in F[T] such that

Question

Let F be a field. Recall that if f(T) and g(T) are polynomials in F[T] such that gcd(f(T), g(T))=d(T), then the Euclidian algorithm computes d(T) (as the last non-zero remainder when dividing f(T) by g(T), g(T) by the remainder, etc). One can prove by going backwards in the Euclidian algorithm that there exist polynomials a(T) and b(T) in F[T] such that a(T)f(T)+b(T)g(T)=d(T).

Work this out in the following specific examples (namely, apply the Euclidian algorithm to find the gcd, then find a(T), b(T) as above):

(a) For f(T)=T^3-1 and g(T)=T^4-1 in Q[T] (rational coefficients).

(b) f(T)=T^3-1 and g(T)=T^5-1 in Q[T] (rational coefficients).

Explanation / Answer

a)T^4-1=(T^3-1)T+T-1

T^3-1=(T-1)(T^2+T+1)

So T-1 is the gcd of T^4-1 and T^3-1 because the remainder is 0 in the last equation

From first equation we get

T-1=(T^4-1)-T(T^3-1)

so a(T)=1,B(T)=-T

b)T^5-1=(T^3-1)T^2+T^2-1

T^3-1=(T^2-1)T+T-1

T^2-1=(T-1)(T+1)

so again T-1 is the gcd of T^5-1 and T^3-1

from second equation we get

T-1=T^3-1-(T^2-1)T

and from frist T^2-1=T^5-1-(T^3-1)T^2

Putting them together

T-1=T^3-1-(T^5-1-(T^3-1)T^2)T=-(T^5-1)T+T^3-1+(T^3-1)T^3=

=-(T^5-1)T+(T^3-1)(T^3+1)

So a(T)=-T,b(T)=T^3+1

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