Show that $\{\sum_{i=N}^{\infty} a_i x^i\}$ forms a Field where $N$ is an Arbitrary Integer and $a_i \in \mathbb{Z}/p\mathbb{Z}$

Asked Aug 01 '24 at 02:41

Active Aug 01 '24 at 02:41

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I am trying to construct an infinite field with finite characteristic. This ring of polynomials $T=\{\sum_{i=N}^{\infty} a_i x^i: a_i \in \mathbb{Z}/p\mathbb{Z}, N\in \mathbb{Z}\}$ where $N$ is possibly negative should do as long as we show that all nonzero elements in $T$ are invertible. How should I go about proving that?

asked Aug 01 '24 at 02:41

Hyakutake

What have you done so far? – Bowei Tang Aug 01 '24 at 02:43
Cf. this question – J. W. Tanner Aug 01 '24 at 02:48
Hint: It is still true if you replace $\mathbb Z/p\mathbb Z$ with any field $F.$ – Thomas Andrews Aug 01 '24 at 02:50
This is the ring of formal Laurent series, not polynomials. An easier construction is to take the rational functions $\mathbb{F}_p(x)$. – Qiaochu Yuan Aug 01 '24 at 05:36

Show that $\{\sum_{i=N}^{\infty} a_i x^i\}$ forms a Field where $N$ is an Arbitrary Integer and $a_i \in \mathbb{Z}/p\mathbb{Z}$

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