ApCoCoALib:RingF16: Difference between revisions

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ApCoCoALib  contains an implementation of the field <math>\mathbb{F}_{16}</math>
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>. A more detailed description of this behavior can be found in the article [[ApCoCoA:Representation_of_finite_fields]].


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

Revision as of 10:32, 6 March 2008

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ApCoCoALib contains an implementation of the field 𝔽16 The field is constructed via 𝔽16=𝔽[x]/(x4+x3+1). 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. x3+x+123+21+20=8+2+1=11. A more detailed description of this behavior can be found in the article ApCoCoA:Representation_of_finite_fields.

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