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Residue Field

In a local ring R, there is a unique maximal ideal m. The corresponding quotient ring R/m is a field called the residue field of R (Matsumura 1986).

For example, the ring Z_p of p-adic integers has maximal ideal pZ_p and residue field

Z_p/pZ_p=F_p,

the finite field with p elements (Cassels 1986).

See also

Algebraic Geometry, Algebraic Number Theory, Finite Field, Local Ring, p-adic Integer, Valuation Ring

Portions of this entry contributed by Todd Rowland

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References

Cassels, J. W. S. "p-adic Fields." Ch. 8 in Local Fields. Cambridge, England: Cambridge University Press, pp. 144-164, 1986.Matsumura, H. "Commutative Rings and Modules." Ch. 1 in Commutative Ring Theory. Cambridge, England: Cambridge University Press, pp. 1-19, 1986.

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Residue Field

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Residue Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ResidueField.html

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