Polynomials over Algebraic Number Fields
Functions like
Factor usually assume that all coefficients in the polynomials they produce must involve only rational numbers. But by setting the option
Extension you can extend the domain of coefficients that will be allowed.
| Factor[poly,Extension->{a1,a2,...}] |
| factor poly allowing coefficients that are rational combinations of the ai |
Factoring polynomials over algebraic number fields.
Allowing only rational number coefficients, this polynomial cannot be factored.
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With coefficients that can involve  , the polynomial can now be factored.
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The polynomial can also be factored if one allows coefficients involving  .
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If one allows coefficients that involve both  and  the polynomial can be factored completely.
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Expand gives the original polynomial back again.
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Factoring polynomials with algebraic number coefficients.
Here is a polynomial with a coefficient involving  .
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By default, Factor will not factor this polynomial.
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But now the field of coefficients is extended by including  , and the polynomial is factored.
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Other polynomial functions work much like
Factor. By default, they treat algebraic number coefficients just like independent symbolic variables. But with the option
Extension->Automatic they perform operations on these coefficients.
By default, Cancel does not reduce these polynomials.
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