If F is a field than does it imply it must also be a Euclidean domain?

[pivoxa15]

Homework Statement

If F is a field than does it imply it must also be a Euclidean domain?

The Attempt at a Solution

Yes since for any a,b in F. a=bq for some q in R. In fact let q=(b^-1)a. So the remainder which occurs in a ED is always 0. So the rule for being a ED is satisfied in any field.

 

[matt grime]

Your answer starts with a minor error: a=1 and b=0 for instance contradict it.

 

[pivoxa15]

I should have added for any a,b not equal to 0.