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:
“Kids won’t come out and thank you each and every time you make a decision they aren’t totally fond of....But in their hearts kids know you’re doing your job, just like they are doing their job by arguing.”
—Fred G. Gosman (20th century)
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—Feodor Dostoyevsky (1821–1881)
“Computers are good at swift, accurate computation and at storing great masses of information. The brain, on the other hand, is not as efficient a number cruncher and its memory is often highly fallible; a basic inexactness is built into its design. The brain’s strong point is its flexibility. It is unsurpassed at making shrewd guesses and at grasping the total meaning of information presented to it.”
—Jeremy Campbell (b. 1931)
“What though the field be lost?
All is not lost; the unconquerable Will,
And study of revenge, immortal hate,
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And what is else not to be overcome?”
—John Milton (1608–1674)