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FrobeniusAutomorphism
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FrobeniusAutomorphism

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gives the value of the Frobenius automorphism at the finite field element a.

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gives the value of the kth functional power of the Frobenius automorphism at a.

Details

  • For a finite field with characteristic , the Frobenius automorphism is given by .
  • All finite field automorphisms are functional powers of the Frobenius automorphism.
  • The number of different field automorphisms of is equal to the extension degree of over .
  • Any field automorphism satisfies equations and .
  • If n is the degree of the MinimalPolynomial f of an element a of , then Table[FrobeniusAutomorphism[a,k],{k,n}] gives all the roots of f in .

Examples

open allclose all

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

Represent a finite field with characteristic and extension degree :

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-o809yt
Out[1]=1

Compute the value of the Frobenius automorphism at an element of the field:

In[2]:=2
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-gwq297
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-djcrnz
Out[3]=3

The third functional power of the Frobenius automorphism:

In[4]:=4
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-vpp8c
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-7ay06
Out[5]=5

Scope  (1)Survey of the scope of standard use cases

Compute the value of the Frobenius automorphism at an element of a finite field:

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-cx03mi
Out[3]=3

Compute all conjugates of a:

In[4]:=4
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-l52b8q
Out[4]=4

Conjugates of a are roots of the minimal polynomial of a:

In[5]:=5
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-kiesl
Out[5]=5

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

Compute the minimal polynomial of an element of a finite field:

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-1pb6k
Out[2]=2

The minimal polynomial of is the product of over all conjugates of :

In[3]:=3
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-dcabog
Out[3]=3

Convert to integer coefficients:

In[4]:=4
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-ngey2
Out[4]=4

Compare with the result obtained using the built-in MinimalPolynomial:

In[5]:=5
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-48d8j
Out[5]=5

Properties & Relations  (5)Properties of the function, and connections to other functions

Frobenius automorphism is a field automorphism:

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-d33o2a
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-ehms77
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-gvurha
Out[4]=4

For a finite field with characteristic , the Frobenius automorphism is given by :

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-gn6eg5
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-voiz9
Out[2]=2

All finite field automorphisms are functional powers of the Frobenius automorphism:

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-i1siun
Out[1]=1

Use FiniteFieldEmbedding to find an automorphism of :

In[2]:=2
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-bwlb2j
Out[2]=2

Identify the functional power of the Frobenius automorphism that gives the same mapping:

In[3]:=3
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-n4s2fm
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-eul20t
Out[4]=4

The number of different field automorphisms of is equal to the extension degree of over :

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-qa4ju
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-buhqp2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-pyk0vz
Out[3]=3

Compute all conjugates of a finite field element a:

In[1]:=1
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-carevw
Out[1]=1

The absolute trace of a is equal to the sum of conjugates:

In[2]:=2
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-kxn2o7
Out[2]=2

Use FiniteFieldElementTrace to compute the absolute trace:

In[3]:=3
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-c14e4b
Out[3]=3

The absolute norm of a is equal to the product of conjugates:

In[4]:=4
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-ubopn
Out[4]=4

Use FiniteFieldElementNorm to compute the absolute norm:

In[5]:=5
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-z1ad7
Out[5]=5

The conjugates are roots of MinimalPolynomial[a]:

In[6]:=6
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https://wolfram.com/xid/0vu3wx0nj254f057ce0-bgtnzk
Out[6]=6
Wolfram Research (2023), FrobeniusAutomorphism, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.
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Wolfram Research (2023), FrobeniusAutomorphism, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

Text

Wolfram Research (2023), FrobeniusAutomorphism, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

✖
Wolfram Research (2023), FrobeniusAutomorphism, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

CMS

Wolfram Language. 2023. "FrobeniusAutomorphism." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

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Wolfram Language. 2023. "FrobeniusAutomorphism." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

APA

Wolfram Language. (2023). FrobeniusAutomorphism. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html

✖
Wolfram Language. (2023). FrobeniusAutomorphism. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html

BibTeX

@misc{reference.wolfram_2025_frobeniusautomorphism, author="Wolfram Research", title="{FrobeniusAutomorphism}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html}", note=[Accessed: 04-April-2025 ]}

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@misc{reference.wolfram_2025_frobeniusautomorphism, author="Wolfram Research", title="{FrobeniusAutomorphism}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_frobeniusautomorphism, organization={Wolfram Research}, title={FrobeniusAutomorphism}, year={2023}, url={https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html}, note=[Accessed: 04-April-2025 ]}

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@online{reference.wolfram_2025_frobeniusautomorphism, organization={Wolfram Research}, title={FrobeniusAutomorphism}, year={2023}, url={https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html}, note=[Accessed: 04-April-2025 ]}