# Thread: Z is not a field.

1. ## Z is not a field.

Why is $\mathbb{Z}$ not a field?

2. Originally Posted by VonNemo19
Why is $\mathbb{Z}$ not a field?
A field requires that every nonzero element has an inverse. The only invertable element of the integers are plus or minus one.

For example $2$ is not invertible in $\mathbb{Z}$

$2x=1$ does not have any integer solutions as $\frac{1}{2} \notin \mathbb{Z}$

3. Oh, OK. I was looking and looking at the definition of a field and comparing it to the properties of the integers and I couldn't find the missing ingredient. So, this little property of the integers, namely that of not having a multiplicitive inverse, is the only condition of the definition of a field that is not satitsfied, correct?

4. Yes.

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# (z, ,•) is field

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