Let D be a Euclidean domain and Q its field of quotients. Furthermore,
let f be the degree function. If a and b are non-zero elements of D, then
a/b = A + r/b
with A and r elements of D such that r = 0 or f(r) < f(b). Decompose b
into a product of powers of a prime
b = p^mq^ns^o etc.
Let b_p = b/p^m, b_q = b/q^n, b_s = b/s^0, etc. Then b_p, b_q, b_o, etc.
are relatively prime. Hence there exist u_p, u_q, u_s, etc. such that
1 = u_pb_p + u_qb_q + u_sb_s + etc.
This yields
a/b = A + ru_p/p^m + ru_q/q^n + ru_s/s^o + etc.
Division with remainder yields v_p, v_q, v_s, etc. such that v_p = 0 or
f(v_p) < f(p^m), etc. and an A' such that
(1) a/b = A' + v_p/p^m + v_q/q^n + v_s/p^s + etc.
Developing v_p p-adically, v_q q-adically, v_s s-adically, etc. one gets
a partial fraction decomposition of a/b, i. e.,
(2) a/b = A' + v_p1/p + v_p2/p^2 + ... + v_pm/p^m + v_q1/q^1 + ...
+ v_qn/q^n + etc.
with v_pi = 0 or f(v_pi) < f(p) for all i, etc.
Partial fractions of type (1) are called partial fractions of the first
kind and partial fractions of type (2) are called to be of the second
kind. When integrating rational functions, we are considering partial
fractions of the second type.
There is a theorem by H. Ostmann saying that in the Euclidean domain D
the development of a/b into a partial fraction of the first kind is
unique if, and only if, D is isomorphic to the ring of polynomials in
one indeterminate over a commutative field. This was generalized to
non-commutative rings with a right-Euclidean algorithm by Heinz Joerg
Claus, Ueber die Partialbruchzerlegung in nicht notwendig kommutativen
euklidischen Ringen. Journal fuer die reine und angewandte Mathematik
194, 88-100, 1955. The paper by Ostmann is also in this journal, vol.
188, 150-161, 1950.
Gauss used the development of fractions of integers into partial fractions
in his computations. See e. g. his disquisitiones, artt. 309-318 and 336.
Since Walter Felscher will read this letter, I should mention that he
drew my attention to the paper by Claus twenty-one years ago.
Best regards
Heinz Lueneburg