NumberFieldSignature
gives the signature of the field 
generated by the algebraic number 
.
Details
- NumberFieldSignature[a] gives a list
of the number of real roots and the number of pairs of conjugate roots for the minimal polynomial of 
.
Examples
open all close allBasic Examples (2)
Scope (4)
NumberFieldSignature[Sqrt[1 + Sqrt[2]]]Root objects:
NumberFieldSignature[Root[1 - 4# ^ 2 + 4# ^ 4 - # ^ 6 + # ^ 8&, 1]]AlgebraicNumber objects:
NumberFieldSignature[AlgebraicNumber[Root[-2 + # ^ 3&, 1], {1, 2, 3}]]NumberFieldSignature threads automatically over lists:
NumberFieldSignature[{Sqrt[2], E ^ (I Pi / 5)}]Applications (1)
Properties & Relations (3)
The minimal polynomial of 
has two real roots and a pair of complex roots:
a = Sqrt[1 + Sqrt[2]];p = MinimalPolynomial[a, x]NSolve[p == 0, x]The signature of the number field generated by 
:
NumberFieldSignature[a]{s, t} = NumberFieldSignature[2 ^ (1 / 3)]s + 2tUse Exponent and MinimalPolynomial to verify the result:
Exponent[MinimalPolynomial[2 ^ (1 / 3), x], x]Find the signature of the number field 
:
primitive = ToNumberField[{Sqrt[2], I}, All][[1, 1]];NumberFieldSignature[primitive]Text
Wolfram Research (2007), NumberFieldSignature, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldSignature.html.
CMS
Wolfram Language. 2007. "NumberFieldSignature." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberFieldSignature.html.
APA
Wolfram Language. (2007). NumberFieldSignature. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberFieldSignature.html