In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic fields, and, more generally, CM fields.
Any number field that is Galois over the rationals must be either totally real or totally imaginary.
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In algebraic number theory, a numberfield is called totallyimaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples...
class number one. The next totally real cubic field (of discriminant 1957) has class number two. The polynomials defining the totally real cubic fields that...
algebraic numberfield Ramification (mathematics) Root of unity Gaussian period Fermat's Last Theorem Class number problem for imaginary quadratic fields Stark–Heegner...
a numberfield. Its elements are elements of the form a + b i {\displaystyle a+bi} where both a and b are rational numbers and i is the imaginary unit...
conjecture is known for a totallyimaginarynumberfield (which has cohomological dimension 2). More generally, for any numberfield k, Martin Kneser, Günter...
in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of −1. The real numbers include the rational...
of the propagation constant from real to imaginary as the frequency drops below the cut-off frequency totally changes the physical nature of the result...
\mathbb {Q} } are called algebraic numberfields, and the algebraic closure of Q {\displaystyle \mathbb {Q} } is the field of algebraic numbers. In mathematical...
number is called an imaginarynumber or is referred to as purely imaginary; if the imaginary part is 0, then the number is a real number. Thus the real numbers...
that tachyonic fields merely give rise to instabilities, not causality violations. The term tachyonic field refers to imaginary mass fields rather than to...
Kroneckerian field A totally real algebraic numberfield or a totallyimaginary quadratic extension of a totally real field. CM-field or J-field An algebraic...
be ordered since the square of the imaginary unit i is −1 (which is negative in any ordered field). Finite fields cannot be ordered. Historically, the...
e. the rest mass m must be imaginary, as a pure imaginarynumber divided by another pure imaginarynumber is a real number. Mass versus weight Effective...
consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the numberfield Q ( i ) {\displaystyle \mathbb {Q} (i)}...
electric and magnetic fields extend well into the substance. In either case the penetration depth is found directly from the imaginary part of the material's...
to the field, a phenomenon known as Larmor precession. Quantum number Azimuthal quantum number Principal quantum number Spin quantum numberTotal angular...
who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one. The Gross–Zagier theorem (Gross & Zagier 1986) describes...
electromagnetic wave, the electric field E has the form where Ek is the (constant) complex amplitude vector, i is the imaginary unit, k is the wave vector...
real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane...
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