# 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```

GAP 3.4.4
April 1997