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Characteristic (Field)

For a Field $K$ with multiplicative identity 1, consider the numbers $2=1+1$, $3=1+1+1$, $4=1+1+1+1$, etc. Either these numbers are all different, in which case we say that $K$ has characteristic 0, or two of them will be equal. In this case, it is straightforward to show that, for some number $p$, we have \(\underbrace{1+1+\ldots+1}_{p{\rm\
times}} = 0\). If $p$ is chosen to be as small as possible, then $p$ will be a Prime, and we say that $K$ has characteristic $p$. The Fields $\Bbb{Q}$, $\Bbb{R}$, $\Bbb{C}$, and the p-adic Number $\Bbb{Q}_p$ have characteristic 0. For $p$ a Prime, the Galois Field GF($p^n$) has characteristic $p$.

If $H$ is a Subfield of $K$, then $H$ and $K$ have the same characteristic.

See also Field, Subfield

© 1996-9 Eric W. Weisstein