The Euclidean['right'](Poly1, Poly2, A) or Euclidean(Poly1, Poly2, A) calling sequence returns a list [m, S] where m is a positive integer and S is an array with m nonzero Ore polynomials such that:

In addition, Remainder['right'](S[m-1], S[m], A) = 0. S is called the right Euclidean polynomial remainder sequence of Poly1 and Poly2.

If the fourth argument c1 of Euclidean['right'] or Euclidean is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and c1[m+1] Poly1 is a least common left multiple (LCLM) of Poly1 and Poly2.

If the fifth argument c2 of Euclidean['right'] or Euclidean is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and  is an LCLM of Poly1 and Poly2.

The Euclidean['left'](Poly1, Poly2, A) calling sequence returns a list [m, S] where m is a positive integer and S is an array with m nonzero Ore polynomials such that:

In addition, . S is called the left Euclidean polynomial remainder sequence of Poly1 and Poly2.

If the fourth argument c1 of Euclidean['left'] is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and Poly1 c1[m+1] is a least common right multiple (LCRM) of Poly1 and Poly2.

If the fifth argument c2 of Euclidean['left'] is is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and Poly1 c1[m+1]= - Poly2 c2[m+1] is an LCRM of Poly1 and Poly2.