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Find the units in $\mathbb Z_4[x]$.

I saw something online about this possibly having an infinite amount of units, but am not sure.

Any help someone can give would be greatly appreciated. Thank you.

m2016
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    See https://math.stackexchange.com/questions/466434/find-a-polynomial-of-degree-0-in-z-4x-that-is-a-unit?rq=1 – Jef May 04 '16 at 18:32

1 Answers1

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The units in $\mathbb{Z}_n$ are exactly those numbers that are relatively prime to $n$. This is because if a number is relatively prime to $n$, then the number will have

$$ \gcd(k, n) = 1 $$

Hence

$$ \exists a, b \in \mathbb{Z}_4 \ \text{such that}\\ k \times a + n \times b = 1 \\ $$

Therefore

$$ k \times a + 0 = 1\mod n \\ \text{where} \ \ k, a \in \mathbb{Z}_4 $$

So, such a number will always have units.

The numers that are relatively primt to $4$ are $\{1, 3\}$. So, there are $2$ numbers that are units to $\mathbb{Z}_4$

Siddharth Bhat
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  • I think the question is asking for the units in the {\it polynomial} ring ${\mathbb {Z}}_{4}}[x]$. Your answer doesn't quite make it I am afraid. The question is answered elsewhere on this website. – student Jan 24 '18 at 03:42