Consider Gauss result :
Gauss's complex multiplication algorithm
Complex multiplication normally involves four multiplications and two additions.
By 1805 Gauss had discovered a way of reducing the number of multiplications to three.
The product$ (a + bi) · (c + di) $can be calculated in the following way.
$k1 = c · (a + b)$
$k2 = a · (d − c)$
$k3 = b · (c + d)$
Real part $= k1 − k3$
Imaginary part $ = k1 + k2.$ This algorithm uses only three multiplications, rather than four, and five additions or subtractions rather than two. If a multiply is more expensive than three adds or subtracts, as when calculating by hand, then there is a gain in speed. On modern computers a multiply and an add can take about the same time so there may be no speed gain.
Now the natural questions :
1) How to extend these results to other abelian rings ?
2) How to prove that the number of multiplications is the lowest possible for a given abelian ring ?
3) how many methods are there for a given abelian ring ?
And perhaps a weird one
4) is it possible to INCREASE the number of multiplications ? ( say an algoritm that uses 5 products for complex number multiplication ).
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5) is this related to the property of Unique factorization ( or non-Unique factorization ) of the considered ring ??
This question is dedicated to my teacher tommy1729.