Hello!

I need a little help understanding this proof. I'm new to this and I'm a little slow =/ I've added questions in italics next to the two places I'm unsure about.

Let F be a field. There is an element 0 $\displaystyle \epsilon$ F such that a + 0 = a for all a $\displaystyle \epsilon$ F. Assume that z $\displaystyle \epsilon$ F is such that a + z = a for all a $\displaystyle \epsilon$ F. Show that z = 0.

Solution:

Suppose z $\displaystyle \epsilon$ F satisfies a + z = a for all a $\displaystyle \epsilon$ F.

Then in particular 0 + z = 0. $\displaystyle \leftarrow$so we're simply using a + z = a from the previous line and putting a = 0 here to get 0 + z = 0?

By the commutative law for addition 0 + z = z + 0.

As a + 0 = a for all a $\displaystyle \epsilon$ F by definition of 0 we moreover have z + 0 = z. $\displaystyle \leftarrow$I don't see how z + 0 = z can follow from a + 0 = a...I thought z + 0 = z was simply one of the axioms of a field (the neutral element for addition). I.e., I don't see how we have used 'a + 0 = a' at all in this proof.

Combining everything we get 0 = 0 + z = z + 0 = z as required.

Thank you!