NumberFieldSignature

NumberFieldSignature[a]

gives the signature of the field generated by the algebraic number a.

Details

  • NumberFieldSignature[a] gives a list {s,t} of the number of real roots and the number of pairs of conjugate roots for the minimal polynomial of a.

Examples

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Basic Examples  (2)

Find the signature of the number field :

A number field with signature :

Scope  (4)

Radical expressions:

Root objects:

AlgebraicNumber objects:

NumberFieldSignature threads automatically over lists:

Applications  (1)

Signatures of Galois extension fields of are of the form or for some integer :

The number field is not a Galois extension of :

Properties & Relations  (3)

The minimal polynomial of has two real roots and a pair of complex roots:

The signature of the number field generated by :

The degree of a number field:

Use Exponent and MinimalPolynomial to verify the result:

Find the signature of the number field :

Wolfram Research (2007), NumberFieldSignature, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldSignature.html.

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

BibTeX

@misc{reference.wolfram_2025_numberfieldsignature, author="Wolfram Research", title="{NumberFieldSignature}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NumberFieldSignature.html}", note=[Accessed: 21-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_numberfieldsignature, organization={Wolfram Research}, title={NumberFieldSignature}, year={2007}, url={https://reference.wolfram.com/language/ref/NumberFieldSignature.html}, note=[Accessed: 21-February-2025 ]}