Difference between revisions of "ApCoCoALib:RingF16"

From ApCoCoAWiki
m (ApCoCoALib:F16 moved to ApCoCoALib:RingF16: The c++ class is called RingF16, not F16.)
(update wtr. the implementation of F16.)
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==about==
 
==about==
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ApCoCoA will soon contain an implementation of the field <math>\mathbb{F}_{16}</math>
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ApCoCoALib  contains an implementation of the field <math>\mathbb{F}_{16}</math>
 
The field is constructed via  <math>\mathbb{F}_{16} = \mathbb{F}[x]/(x^4 + x^3 + 1)</math>.  
 
The field is constructed via  <math>\mathbb{F}_{16} = \mathbb{F}[x]/(x^4 + x^3 + 1)</math>.  
 
The field's elements are represented as integers between 0 and 15.  The corresponding mapping is the substitution homomorphism, mapping x to 2. Therefore we have e.g. <math>x^3 + x + 1 \mapsto 2^3 + 2^1 + 2^0 = 8 + 2  + 1 = 11</math>
 
The field's elements are represented as integers between 0 and 15.  The corresponding mapping is the substitution homomorphism, mapping x to 2. Therefore we have e.g. <math>x^3 + x + 1 \mapsto 2^3 + 2^1 + 2^0 = 8 + 2  + 1 = 11</math>
  
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==alternative representations==
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This implementation is contained in the files RingF16,[CH]. A new instance can be created with the command:
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Instead of using <math>x^4 + x^3 + 1 </math>, we could have also chosen another irreducible polynomial of degree 4.
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ApCoCoA::AlgebraicCore::NewRingF16();
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In total, there are three ireducible ones, namely <math>\{x^4 + x^3 + 1,x^4 + x^3 + 1, x^4 + x^3 + x^2 + x + 1 \}</math>
 
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If you have a system, which is based on one of the other irreducibly polynomials, you have to construct an isomorphism between the different representations of the field (which is unique). These isomorphisms can be constructed by mapping the irreducible polynomial's roots to the roots of
 
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x^4 + x^3 + 1, respecting the fields galois-group.
 
  
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The irreducible polynomials and roots are:
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An example, describing how to use RingF16, especially together with ring homomorphisms can be found in ApCoCoALib's example directory. It is named ex-RingF16.C
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[x^4 + x^3 + 1, [x^2, x, x^3 + x^2 + x, x^3 + 1]]
 
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[x^4 + x + 1, [x^2, x, x^2 + 1, x + 1]]
 
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[x^4 + x^3 + x^2 + x + 1, [x^3, x^2, x, x^3 + x^2 + x + 1]]
 
  
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They were produced with the CoCoA4 code, explained [[HowTo:construct_fields|here]].
 
  
 
[[Category:ApCoCoALib_Manual]]
 
[[Category:ApCoCoALib_Manual]]

Revision as of 09:03, 4 February 2008

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about

ApCoCoALib contains an implementation of the field The field is constructed via . The field's elements are represented as integers between 0 and 15. The corresponding mapping is the substitution homomorphism, mapping x to 2. Therefore we have e.g.

This implementation is contained in the files RingF16,[CH]. A new instance can be created with the command: 

ApCoCoA::AlgebraicCore::NewRingF16();

An example, describing how to use RingF16, especially together with ring homomorphisms can be found in ApCoCoALib's example directory. It is named ex-RingF16.C

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