Show that in a field always $0\ne1$ Suppose that $F$ is a field and prove that $0\ne1$

According to the definition of a field I know that the zero element is different from the one element, but is there a scientific proof for that?
In a field, $0\neq 1$ by definition. This is to exclude the zero ring from being a field.

This leads to a natural question which is an easy but illuminating exercise: If we are in a ring with $0=1$, show that every element equals $0$.
There is a principle in proof theory that if you can find a set of variables such that axioms hold but a conclusion does not, then it is impossible to prove the conclusion from the axioms.
For example, suppose you know the following axioms:

$$x &gt; y \tag{A1}$$
$$x &gt; z \tag{A2}$$

Call $P$ a proposition to be proven as $y &gt; z$.  Well, if we let $x = 3, y = 1, z = 2$.  Note that under this assignment, A1 and A2 are true, but $P$ is false.  Therefore you cannot prove $P$ from A1 and A2.

There are 10 or so axioms of field theory.  The one element field satisfies the axioms of field theory.  The one element field also satisfies $0_F = 1_F$, therefore it is impossible to prove $0_F \ne 1_F$ from just field axioms.