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AlgebraicNumberNorm
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AlgebraicNumberNorm

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gives the norm of the algebraic number a.

Details and Options

  • The norm of a is defined to be the product of the roots of its minimal polynomial.
  • AlgebraicNumberNorm[a,Extension->θ] finds the norm of a over the field .

Examples

open allclose all

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

Norms of algebraic numbers:

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-gqg0h1
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn1pdqw9ra-bkcuhg
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0btn1pdqw9ra-5l5ynn
Out[3]=3

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

Integers and rational numbers:

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-7ksa7f
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn1pdqw9ra-0uqmb0
Out[2]=2

Radical expressions:

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-poo44j
Out[1]=1

Root and AlgebraicNumber objects:

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-w3a6kl
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn1pdqw9ra-qve1l5
Out[2]=2

AlgebraicNumberNorm automatically threads over lists:

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-gy6c3t
Out[1]=1

Options  (1)Common values & functionality for each option

Extension  (1)

Norm of over :

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-fmam3e
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

is irreducible in :

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-trkj2c
Out[1]=1

Since AlgebraicNumberNorm is multiplicative, having a prime norm implies the original number is prime:

In[2]:=2
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https://wolfram.com/xid/0btn1pdqw9ra-4t52km
Out[2]=2

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

AlgebraicNumberNorm is multiplicative:

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-3osxzb
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn1pdqw9ra-p99r75
Out[2]=2

Units in a number field have norm :

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-9r13z5
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn1pdqw9ra-u1dc6l
Out[2]=2

Compute the smallest field that includes , i.e. :

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-bvnwy3
Out[1]=1

Compute the norm in that field:

In[2]:=2
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https://wolfram.com/xid/0btn1pdqw9ra-owxq2
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Plot of norms of elements in :

In[1]:=1
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https://wolfram.com/xid/0btn1pdqw9ra-ii94a4
Out[1]=1
Wolfram Research (2007), AlgebraicNumberNorm, Wolfram Language function, https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html.
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Wolfram Research (2007), AlgebraicNumberNorm, Wolfram Language function, https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_algebraicnumbernorm, author="Wolfram Research", title="{AlgebraicNumberNorm}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html}", note=[Accessed: 29-April-2025 ]}

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@misc{reference.wolfram_2025_algebraicnumbernorm, author="Wolfram Research", title="{AlgebraicNumberNorm}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html}", note=[Accessed: 29-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_algebraicnumbernorm, organization={Wolfram Research}, title={AlgebraicNumberNorm}, year={2007}, url={https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html}, note=[Accessed: 29-April-2025 ]}

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@online{reference.wolfram_2025_algebraicnumbernorm, organization={Wolfram Research}, title={AlgebraicNumberNorm}, year={2007}, url={https://reference.wolfram.com/language/ref/AlgebraicNumberNorm.html}, note=[Accessed: 29-April-2025 ]}