Introductory examples:
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All factors in the output will be monic polynomials, even if rationals must be introduced:
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The input will be factored over the variable(s) given by the optional v parameter. Variables not included in v will be treated as constants:
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The output will not necessarily be factored into a product of irreducibles, but partial factorization in the input is preserved, so long as no factors have a GCD greater than 1:
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However, with option 'expanded'=true, factors with the same multiplicity will be combined and expanded:
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| (18) |
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Trigonometric and exponential functions will be converted to an algebraic form, if possible:
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| (20) |
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All algebraic numbers will be reduced modulo their minimal polynomials (see Reduce):
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| (22) |
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Non-algebraic sub-expressions such as will be frozen and replaced by new variables which are not treated as constants if optional argument v is not included:
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Such functions can also be included in v, but use caution, as some functions may produce strange results. Constants such as cannot be included, as they will be converted to algebraic numbers whenever possible:
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The arguments of functions in the input will be recursively normalized and simplified (see Normal):
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Non-algebraic subexpressions may evaluate to something algebraic after being normalized:
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Algebraic functions such as are not accepted:
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Floats are not accepted.
When a non-indexed RootOf is given in the input, sometimes the square-free factorization can still be computed:
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However, in this case, to compute a factorization, Squarefree must know whether represents or , in order to guarantee that the output is actually square free:
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| (34) |
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To force Squarefree to reduce this RootOf to one of lower degree by picking one of the substitutions arbitrarily, use option 'symbolic'=true. Here, it makes the substitution = :
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Squarefree can factor out un-indexed RootOfs as constants, as long as they don't contain zero divisors:
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| (36) |
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In this case, option 'symbolic'=true will eliminate the zero divisor:
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Using option 'characteristic', the squarefree factorization can per performed over finite fields:
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| (40) |
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| (41) |
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With option 'makeindependent'=true, the input will be checked for algebraic dependencies even if there are more than algebraic objects in the input:
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| (46) |
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| (48) |
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| (49) |
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With option 'makeindependent'=false, the input will never be checked for algebraic dependencies:
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| (52) |