# 13.2 Cyclotomic Integers

A cyclotomic is called integral or cyclotomic integer if all coefficients of its minimal polynomial are integers. Since the base used is an integral base (see ZumbroichBase), the subring of cyclotomic integers in a cyclotomic field is formed by those cyclotomics which have not only rational but integral coefficients in their representation as sums of roots of unity. For example, square ATLAS irrationalities), any root of unity is a cyclotomic integer, character values are always cyclotomic integers, but all rationals which are not integers are not cyclotomic integers. (See IsCycInt)

```    gap> ER( 5 );                # The square root of 5 is a cyclotomic
E(5)-E(5)^2-E(5)^3+E(5)^4    # integer, it has integral coefficients.
gap> 1/2 * ER( 5 );          # This is not a cyclotomic integer, \ldots
1/2*E(5)-1/2*E(5)^2-1/2*E(5)^3+1/2*E(5)^4
gap> 1/2 * ER( 5 ) - 1/2;    # \ldots but this is one.
E(5)+E(5)^4```

GAP 3.4.4
April 1997