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The field of rationals is the set of rational numbers, which form a field. This field is commonly denoted Q (doublestruck Q).

The field of reals is the set of real numbers, which form a field. This field is commonly denoted R (doublestruck R).

The characteristic exponent of a field is 1 if the field characteristic is 0 and p if the field characteristic is p.

A field which is complete with respect to a discrete valuation is called a local field if its field of residue classes is finite. The Hasse principle is one of the chief ...

A quadratic field Q(sqrt(D)) with D>0.

A field K is said to be an extension field (or field extension, or extension), denoted K/F, of a field F if F is a subfield of K. For example, the complex numbers are an ...

In a local ring R, there is only one maximal ideal m. Hence, R has only one quotient ring R/m which is a field. This field is called the residue field.

The order of a finite field is the number of elements it contains.

A vector field v for which the curl vanishes, del xv=0.

A place nu of a number field k is an isomorphism class of field maps k onto a dense subfield of a nondiscrete locally compact field k_nu. In the function field case, let F be ...