- #1

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- TL;DR Summary
- is ##\mathbb{Z} = \mathbb{N}[[\sqrt 1]] / \{0\}##?

Sometimes I have seen a process to build integers and rationals via a sort of Grothendieck product, Z being classes of equivalence in N x N, and Q being classes of equivalence in Z x Z.

Now, I was wondering if it makes sense to consider the integers as the extension of ##\mathbb{N}## by ##\sqrt 1##, so that an element of ##\mathbb{N}[[\sqrt 1]]## is generically of the form ##n + m \sqrt 1##, and then ##\mathbb{Z}## is the quotient by the ideal ##n + n \sqrt 1##. How far fetched (or plainly wrong) is this?

Now, I was wondering if it makes sense to consider the integers as the extension of ##\mathbb{N}## by ##\sqrt 1##, so that an element of ##\mathbb{N}[[\sqrt 1]]## is generically of the form ##n + m \sqrt 1##, and then ##\mathbb{Z}## is the quotient by the ideal ##n + n \sqrt 1##. How far fetched (or plainly wrong) is this?