In number theory, a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers. Equivalent conditions are that K is generated over Q by one root of an integer polynomial P, all of the roots of P being real; or that the tensor product algebra of K with the real field, over Q, is a product of copies of R.
For example, quadratic fields K of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial P irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will not be totally real, although it is a field of real numbers.
The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two.
Any number field that is Galois over the rationals must be either totally real or totally imaginary.
Famous quotes containing the words totally, real, number and/or field:
“On our streets it is the sight of a totally unknown face or figure which arrests the attention, rather than, as in big cities, the strangeness of occasionally seeing someone you know.”
—For the State of Vermont, U.S. public relief program (1935-1943)
“Patience, to hear frivolous, impertinent, and unreasonable applications: with address enough to refuse, without offending; or, by your manner of granting, to double the obligation: dexterity enough to conceal a truth, without telling a lie: sagacity enough to read other people’s countenances: and serenity enough not to let them discover anything by yours; a seeming frankness, with a real reserve. These are the rudiments of a politician; the world must be your grammar.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“... in every State there are more women who can read and write than the whole number of illiterate male voters; more white women who can read and write than all Negro voters; more American women who can read and write than all foreign voters.”
—National Woman Suffrage Association. As quoted in History of Woman Suffrage, vol. 4, ch. 13, by Susan B. Anthony and Ida Husted Harper (1902)
“The head must bow, and the back will have to bend,
Wherever the darkey may go;
A few more days, and the trouble all will end,
In the field where the sugar-canes grow.
A few more days for to tote the weary load,—
No matter, ‘t will never be light;
A few more days till we totter on the road:—
Then my old Kentucky home, good-night!”
—Stephen Collins Foster (1826–1884)