Establish a [itex]\bullet[/itex] 0 = 0

So you want to prove :

a0 = 0 in some field F.

So I'm presuming you're allowed to assume the existence of a unique zero element and unique additive inverses.

Then :

a0 = 0 + a0
= (-(a0) + a0) + a0
.....

As much as id like to help you more, the rest should be obvious. Just use the axioms to justify your steps.

 

A similar approach is to use that 0=0+0.

In this attempt, you haven't made it clear how the first equality implies the second.

 

[itex]\bullet[/itex] is a label for the operation of multiplication in a field F. There is an operation of addition in the field, for which the following is true:
[tex]
x = x + 0, \forall x \in F
[/tex]
Multiply by a, and use the distributive law:
[tex]
a \bullet x = a \bullet (x + 0)
[/tex]
[tex]
a \bullet x = a \bullet x + a \bullet 0
[/tex]
Use the group properties of addition of the field. What can you say about [itex]a \bullet 0[/itex] then? Similarly, you can multiply from the left and draw a conclusion for [itex]0 \bullet a[/itex].