Editing Totally real number field (section)

[[File:TotallyReal.svg|225px|thumb|right|The number field '''Q'''(√2) sits inside '''R''', and the two embeddings of the field into '''C''' send every element in the field to another element of '''R''', hence the field is totally real.]]

In [[number theory]], a [[number field]] ''F'' is called '''totally real''' if for each [[embedding]] of ''F'' into the [[complex number]]s the [[image (mathematics)|image]] lies inside the [[real number]]s. Equivalent conditions are that ''F'' is generated over '''Q''' by one [[zero of a function|root]] of an [[integer polynomial]] ''P'', all of the roots of ''P'' being real; or that the [[tensor product of fields|tensor product algebra]] of ''F'' with the real field, over '''Q''', is [[isomorphic]] to a tensor power of '''R'''.

For example, [[quadratic field]]s ''F'' 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 field]]s, a cubic integer polynomial ''P'' [[irreducible polynomial|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 (mathematics)|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 (mathematics)|subfield]] over which it has degree two.

Any number field that is [[Galois extension|Galois]] over the [[rational number|rationals]] must be either totally real or [[totally imaginary number field|totally imaginary]].