Proof
(related to Proposition: In a Field, $0$ Is Unequal $1$)
 By hypothesis, $(F,+,\cdot)$ is a field.
 Suppose, $0=1$.
 Take $x\in F$ with $x\neq 0$^{1}.
 Because $0$ is the neutral element of addition in $F$, $1$ is its additive inverse, i.e. $10=0.$
 From the distributivity laws in $F,$ we get $x\cdot 0=x\cdot (10)=x\cdot 1x\cdot 0.$
 Because $1$ is the neutral element of multiplication in $F$, it follows $x\cdot 0=xx\cdot 0,$ or $x=x\cdot 0+x\cdot 0.$
 From the distributivity laws, it follows $x=x\cdot (0+0),$ and, since $0$ is neutral with respect to the an addition, $x=x\cdot 0.$
 Altogether, it follows $x=xx,$ or $x=0$, in contradiction to $x\neq 0.$
 The assumption $0=1$ must be false, i.e. $0\neq 1.$
∎
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References
Bibliography
 Knauer Ulrich: "Diskrete Strukturen  kurz gefasst", Spektrum Akademischer Verlag, 2001
Footnotes