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## Homework Statement

Through my exploration/experimentation, I came up with this little conjecture: Let ##F## be a field and ##x## some element in the field. If there exists a natural number ##m## such that ##x^m = 0##, then ##x=0##. In other words, a field contains no nontrivial nilpotent elements.

## Homework Equations

## The Attempt at a Solution

Suppose that such a natural number ##m## exists for which ##x^m = 0##, or ##x \cdot x^{m-1}##. Since fields do not contain zero divisors, either ##x=0## or ##x^{m-1}##. If the latter holds, we are finished, so assume ##x^{m-1} = 0## or ##x \cdot x^{m-1} =0##. This implies ##x^{m-2} = 0##, etc.

It seems that I could repeat this process until I obtain ##x=0##, suggesting that the conjecture is true (unless I made some elementary mistake, which is not unlikely!). However, is there anyway of making this process/argument more rigorous?