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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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