Glossary of Field Theory - Definition of A Field

Definition of A Field

A field is a commutative ring (F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division.

The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F×;

The ring of polynomials in the variable x with coefficients in F is denoted by F.

Read more about this topic:  Glossary Of Field Theory

Famous quotes containing the words definition of a, definition of, definition and/or field:

    ... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.
    George Eliot [Mary Ann (or Marian)

    It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.
    François, Duc De La Rochefoucauld (1613–1680)

    It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.
    François, Duc De La Rochefoucauld (1613–1680)

    The birds their quire apply; airs, vernal airs,
    Breathing the smell of field and grove, attune
    The trembling leaves, while universal Pan,
    Knit with the Graces and the Hours in dance,
    Led on th’ eternal Spring.
    John Milton (1608–1674)