Fields of power series
• Power series expansions
By analogy with the decimal expansion for a real
number, we introduce the following formal expression :
X = anTn + an-1Tn-1
+ an-2Tn-2 +......
Here T is a purely formal symbol,
regarded as a variable. This expression is a power series expansion. The
ai are called the coefficients
of the expansion and the exponents of the variable
T are rational integers. We will see below what these coefficients
can be. As in the case of a decimal expansion, the mutiplicative symbol between the coefficients and a power of the variable are omitted. The expansion may be finite or infinite. The expansion is zero if all
the coefficients are zero. If not all coefficients vanish, then the first term
of the expansion has a non-zero coefficient. In this case, the exponent of
T in this term is a positive or negative integer, or zero. This rational
integer n is called the degree
of the expansion. We call such an expression a formal number.
• Formal Integers
If the above expansion has only a finite number
of terms, each one with a positive degree, we can write
X = anTn + an-1Tn-1
+ .......+ amTm
where n >= m >= 0 . This expression
is known to be a polynomial in T
of degree n . We say that we have
a formal integer. For a general formal number, we use the same terminology as
in the case of real numbers. We call the sum of the terms of positive degree the
integral part (eventually zero if the degree of
X is negative)
while the remaining part (if there is any) is the fractional part. If the expansion
X is finite, the last term being
akT k with
a negative degree k < 0 , then
X is the quotient of two polynomials in
T . Indeed we have
X = (anTn-k + an-1Tn-k-1
+ .......+ ak) / T-k
Thus a finite expansion is the quotient of two formal integers, as in the case
of real numbers.
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