Difference between revisions of "ApCoCoALib:RingF16"
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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 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. . A more detailed description of this behavior can be found in the article ApCoCoA:Representation_of_finite_fields.
![{\displaystyle \mathbb {F} _{16}=\mathbb {F} [x]/(x^{4}+x^{3}+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e093fafdc9be0d3a15c99e05b1422f3a293de8d3)
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
