Construct $\mathbb{Z}(X)$

Question

Is there any way to construct the ring $\mathbb{Z}(X)$ in a computer algebra system (e.g. MAGMA, Sage), where the parameter $X$ is adjoined to $\mathbb{Z}$.

(Note that this is not a field, so I cannot use the command for computing function field of the integral domain $\mathbb{Z}[X]$.)

Thanks in advance.

Edit

Thanks to henning-makholm and sharding4 for helping to ask correct question.

Answer 1

I dont know how to remove answer or mark it as non-answer (without deleting it)!

The $\mathbb{Z}(X)$ is localization of $\mathbb{Z}[X]$, is absolutely correct. However the underlying multiplicative set is not $\{X,X^2,\ldots\}$, but $\mathbb{Z}$.

When you invert all non-costant polynomials, the ring you would get is $\mathbb{Z}(X)$ .

e.g. the element $\frac{1}{1+X}$ is in the ring and so is $\frac{1}{2+X}$, but $\frac{1}{2}$ is not there.

From computation point of view, I donno! Any pointers?

Hope this clarifies!

-- Mike

Answer 2

The ring you are looking for is the Laurent Polynomial ring over $\mathbb{Z}$, namely $\mathbb{Z}[x,x^{-1}]$. This can be constructed as follows: $$\frac{\mathbb{Z}[x][y]}{\langle xy-1\rangle}$$

This you can compute in almost all CASs.

Hope this helps.

-- Mike