13.9 Comparisons of Cyclotomics

To compare cyclotomics, the operators <, <=, =, =, and < can be used, the result will be true if the first operand is smaller, smaller or equal, equal, larger or equal, larger, or inequal, respectively, and false otherwise.

Cyclotomics are ordered as follows: The relation between rationals is as usual, and rationals are smaller than irrational cyclotomics. For two irrational cyclotomics z1, z2 which lie in different minimal cyclotomic fields, we have <z1> < <z2> if and only if 'NofCyc(<z1>)' < 'NofCyc(<z2>)'); if 'NofCyc(<z1>)' = 'NofCyc(<z2>)'), that one is smaller that has the smaller coefficient vector, i.e., <z1> leq <z2> if and only if 'COEFFSCYC(<z1>)' leq 'COEFFSCYC(<z2>)'.

You can compare cyclotomics with objects of other types; all objects which are not cyclotomics are larger than cyclotomics.

    gap> E(5) < E(6);     # the latter value lies in $Q_3$
    false
    gap> E(3) < E(3)^2;    # both lie in $Q_3$, so compare coefficients
    false
    gap> 3 < E(3); E(5) < E(7);
    true
    true
    gap> E(728) < (1,2);
    true

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