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NumberFieldIntegralBasis
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NumberFieldIntegralBasis

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gives an integral basis for the field generated by the algebraic number a.

Details

Examples

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Basic Examples  (1)Summary of the most common use cases

Find an integral basis for the field :

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-fk3fu1
Out[1]=1

Scope  (4)Survey of the scope of standard use cases

Radical objects:

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-ibfl2r
Out[1]=1

Root objects:

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-dgye32
Out[1]=1

AlgebraicNumber objects:

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-i8iaoc
Out[1]=1

NumberFieldIntegralBasis threads over lists:

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-skz3wc
Out[1]=1

Properties & Relations  (3)Properties of the function, and connections to other functions

Elements of an integral basis are all algebraic integers:

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-15zs8h
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bc6ul96ajbldee-vpiwjh
Out[2]=2

Algebraic integers are linear combinations of elements of the integral basis:

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-gigyx4
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bc6ul96ajbldee-9cc9sa
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bc6ul96ajbldee-qffg6g
Out[3]=3

Use ToNumberField to find an integral basis for the field :

In[1]:=1
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https://wolfram.com/xid/0bc6ul96ajbldee-ciug6w
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bc6ul96ajbldee-twksah
Out[2]=2
Wolfram Research (2007), NumberFieldIntegralBasis, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html.
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Wolfram Research (2007), NumberFieldIntegralBasis, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html.

Text

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

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

CMS

Wolfram Language. 2007. "NumberFieldIntegralBasis." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html.

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Wolfram Language. 2007. "NumberFieldIntegralBasis." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html.

APA

Wolfram Language. (2007). NumberFieldIntegralBasis. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html

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Wolfram Language. (2007). NumberFieldIntegralBasis. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html

BibTeX

@misc{reference.wolfram_2025_numberfieldintegralbasis, author="Wolfram Research", title="{NumberFieldIntegralBasis}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html}", note=[Accessed: 22-April-2025 ]}

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@misc{reference.wolfram_2025_numberfieldintegralbasis, author="Wolfram Research", title="{NumberFieldIntegralBasis}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html}", note=[Accessed: 22-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_numberfieldintegralbasis, organization={Wolfram Research}, title={NumberFieldIntegralBasis}, year={2007}, url={https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html}, note=[Accessed: 22-April-2025 ]}

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@online{reference.wolfram_2025_numberfieldintegralbasis, organization={Wolfram Research}, title={NumberFieldIntegralBasis}, year={2007}, url={https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html}, note=[Accessed: 22-April-2025 ]}