Trying to show a set is a field

[Zoe-b]

Homework Statement

I'm doing a question where part of it is to show that the span of the following set over the rationals is a field.
If w is the primitive cube root of unity and z is the positive cube root of 2, the set in question is:

(1,z,z²,w, wz, wz²)

Homework Equations

The Attempt at a Solution

I have already shown this span is closed under multiplication. Almost by definition it is closed under addition/subtraction. I have also shown that each basis element has an inverse in the span. However, when I come to then showing that every element has an inverse, I struggle.. a general element is obviously a linear combination of 6 elements and I can't exactly rationalise the denominator. Any hints at all?

Thanks,
Zoe

 

[mighty2000]

Dear Zoe,

To show that every element in the span has an inverse, you can use the fact that the rational numbers are a field. This means that every non-zero rational number has an inverse in the field. So, for any element in the span that is a non-zero rational number, its inverse will also be in the span.

For elements that are not rational numbers, you can use the fact that the span is closed under multiplication and addition to manipulate the expression and show that it can be written as a rational number multiplied by an element in the span. This will then show that the element has an inverse in the span.

I hope this helps. Good luck with your question!