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Field Characteristic


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 the latter case, it is straightforward to show that, for some number p, we have 1+1+...+1_()_(p 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 characteristic of a field K is sometimes denoted ch(K).

The fields Q (rationals), R (reals), C (complex numbers), and the p-adic numbers Q_p have characteristic 0. For p a prime, the finite field GF(p^n) has characteristic p.

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


See also

Characteristic, Field, Field Characteristic Exponent, Finite Field, Subfield

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References

Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 422, 1998.

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Field Characteristic

Cite this as:

Weisstein, Eric W. "Field Characteristic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FieldCharacteristic.html

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