# Inverse-producing extensions of Topological Algebras/Algebra of polynomials

## Introduction[edit | edit source]

A polynomial algebra is a vector space of polynomials, where the coefficients come from the given algebra . The polynomial algebra is an essential tool to construct an algebra extension of in which a given is invertible if it satisfies certain topological invertibility criteria.

### Remark: Algebra expansion[edit | edit source]

In the construction of algebra extension in which a is invertible, the first step is to consider the algebra of polynomials . The following figure shows how the algebra extension is constructed over the polynomial algebra.

## Algebraic closure[edit | edit source]

We only extend the algebra to contain an additional element , which is to be contained in an algebra extension of . Since multiplication and addition must be completed in , polynomials result from multiplications and with coefficients n, which must be contained as summands as polynomials in the algebra expansion.

### Extension of the algebra[edit | edit source]

This implies the closure of the

- multiplicative linkage of with itself and therefore with must also be in again,
- the arbitrary multiplicative links of with elements from lie again in , i.e. .
- the additive algebraic algebraic closure also eventually requires that additive links from lie again in .

### Polynomials with coefficients from the given algebra[edit | edit source]

Out of this necessity, one considers polynomials with coefficients from as the first step in constructing an algebra expansion in which a can be invertible.

## Polynomial algebra[edit | edit source]

We now consider, for a given topological algebra , the set of polynomials with coefficients in .

and power series with coefficients in algebra .

### Degree of polynomials[edit | edit source]

First of all, polynomials would be more formally notated in the above form with and with would indicate the degree of the polynomial. For the Cauchy product of two polynomials , however, this notation is unsuitable, since in the addition and multiplication two polynomials the handling of the degree entails additional formal overhead, which, however, does not matter for the further considerations of algebra expansions.

### Notation for the polynomial algebra[edit | edit source]

Therefore, polynomials are defined as follows over "finite" sequences , which from an index bound consists only of the zero vector in .

## Cauchy product[edit | edit source]

Given, in general, two polynomials with coefficients from .

Then Cauchy product of is defined as follows:

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