invertible formal power series

Theorem.  Let R be a commutative ring with non-zero unity.  A formal power series

f(X):=i=0aiXi (1)

is invertiblePlanetmathPlanetmath in the ring R[[X]]  iff  a0 is invertible in the ring R.

Proof.1.  Let f(X) have the multiplicative inverseMathworldPlanetmathg(X):=i=0biXi.  Since

f(X)g(X)=i=0j=0iajbi-jXi= 1,

we see that  a0b0=1, i.e. a0 is an invertible element (unit) of R.

2.  Assume conversely that a0 is invertible in R.  For making from a formal power series

g(X):=i=0biXi (2)

the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of f(X)=i=0aiXi, we first choose  b0:=a0-1.  For all already defined coefficients b0,b1,,bi-1 let the next coefficient be defined as


This equation means that


vanishes for all  i=1, 2,;  since  a0b0=1,  the productPlanetmathPlanetmath of the formal power series (1) and (2) becomes simply equal to 1.  Accordingly, f(x) is invertible.

Title invertible formal power series
Canonical name InvertibleFormalPowerSeries
Date of creation 2016-04-27 10:47:14
Last modified on 2016-04-27 10:47:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 13H05
Classification msc 13F25
Classification msc 13J05
Related topic RulesOfCalculusForDerivativeOfFormalPowerSeries