The rate of change of potential is force
But not here

[qoute]
That's simply not true.[/quote]Why not true?The definition of potntial is the work done to bring a unit mass from infinity to a point, and is infinite here since the force is constant all over.In the classical case, force at infinity is zero

The potential difference between any two points will be the work done to move a unit mass between those two points. (In some cases it makes sense to define the potential at infinity equal to zero, but not if the field is everywhere uniform.) Clearly the potential varies along the line of the force.

This is just plain wrong. You can't define the potential at one point to be infinity, then calculate that at another point it's also infinity, so the difference between them is zero. Apart from not being the way to solve the problem, this is mathematically incorrect.

A constant field has a potential growing linearly with distance.

It is mathematically incorrect, the answer is actually uncertain.
I made use of the fact that since everywhere in the field the force on a unit mass is the same, the potential everywhere is the same too, but this does not go well with the math
There is no imbalance in this field, and hence there should be no potential difference.
But this is not real, and hence the answer is not real

I haven't read this thread in it's entirety, but I would like to comment on your last post:

As you say, this does not "go with the math" and is therefore incorrect! If there force on a unit mass is the same everywhere, that doesn't mean that potential is the same! In one dimension, the force is defined thus,

[tex]F = -\frac{dV}{dx}[/tex]

We assume that the force is constant,

[tex]\frac{dV}{dx} = \text{const}[/tex]

The potential is then (denoting the constant c_{1}),

Vanadium said that a lot earlier. And I replied "the potential at a point is the work done to bring a unit mass from infinity to that point and thus the potential everywhere in this field
is infinite"
Read my edited previous post.
There is no real answer to this because the situation itself is not real

Exactly the problem. There is no such place as "at infinity", so trying to calculate the potential "at infinity" is where your logic goes wrong. You can't treat infinity like a real number and try to do maths with it. Think about what's really happening, without bringing infinity into it, and you should be able to make sense of it.

When we say "it takes an infinite amount of work to move to infinity" you can't take that literally. It's a shorthand for saying, "the further you go, the more work is required, without any upper limit".

Don't forget that you can always add a constant to a potential, you don't have to evaluate it "at infinity" to decide what it is elsewhere.

If you have read all my posts,:
Yes, I agree the force is the negative gradient of potential, this is a fact
Here there is a fixed force with an uncertain difference of potential, which is what makes the thing unreal