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• Simplified writing

Following this analogy and since the presence of the indeterminate T is formal, we can use a shorter writing for a formal number. For instance the following formal number of degree five

X = aT5 + bT3+ cT2 + dT-1 + eT-2

can simply be denoted by

X =

Observe that we can also write

X = a0bc00de / 100

• The base field

It is now necessary to say what kind of quantities the coefficients must be. A set of numbers is inseparable from the operations made on these numbers, namely addition and multiplication. When every number has an opposite for the addition in the set and every non-zero number has an inverse for the multiplication also in the set, then the set is called a field. Fields are convenient sets of numbers because the existence of the opposite and of the inverse of a number implies the possibility of substracting (by adding the opposite) and dividing (by multiplying by the inverse). Every field contains in the first place 0 and 1. For instance R and Q are fields, but this is not so for N or Z.
Considering power series as numbers, we naturally want to be able to add them, to multiply them and also to substract and divide them. According to elementary rules on algebraic expressions, we should have

aTn + bTn = (a + b)Tn   and   aTn * bTm = (a * b)Tn+m

-(aTn) = (-a)Tn  and   (aTn)-1 = a-1T-n

This is only made possible if the coefficients belong to a set K with addition, multiplication and where each element has an opposite and an inverse (if not zero) . Therefore K need to be a field. The power series with coefficients in K will be said formal numbers over the base field K.


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