Define a Field

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Pre-Calculus › Define a Field

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Which of the following is not true about a field. (Note: the real numbers $\small \mathbb{R}$ is a field)

For every element $\small a$ in the field, there is another element $\small b$ such that their product $\small \small ab$ is equal to $\small 1$ , where $\small 1$ is the multiplicative identity, $\small 1$ in the case of real numbers.

For every element $\small a$ in the field, there is another element $\small b$ such that their sum $\small a+b$ is equal to $\small 0$ , where $\small 0$ is the additive identity.

There is an element $\small e$ in the field such that $\small e+a=a+e=a$ for any element $\small a$ in the field.

We have $\small ab=ba$ for any $\small a$ and $\small b$ in the field.

A field can be defined in many ways.

Explanation

It is not the case that for any element $\small a$ in a field, there is another one $\small b$ such that their product is $\small 1$ . Take $\small 0$ in the real numbers. Multiply $\small 0$ by any number and you get $\small 0$ , so you will never get $\small 1$ . This is true for any field that has more than 1 element.

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