60.4 Galois Groups of Abelian Number Fields

The field automorphisms of the cyclotomic field ? n subscript ? 𝑛 ?_{n} (see ChapterΒ 18) are given by the linear maps * k absent π‘˜ *k on ? n subscript ? 𝑛 ?_{n} that are defined by E ( n ) * k = superscript 𝑛 absent π‘˜ absent (n)^{{*k}}= E ( n ) k superscript 𝑛 π‘˜ (n)^{k} , where 1 ≀ k ⁒ & ⁒ l ⁒ t ; n 1 π‘˜ & 𝑙 𝑑 𝑛 1\leq k<n and Gcd ( n , k ) = 1 𝑛 π‘˜ 1 (n,k)=1 hold (seeΒ GaloisCyc (18.5-1)). Note that this action is not equal to exponentiation of cyclotomics, i.e., for general cyclotomics z 𝑧 z , z * k superscript 𝑧 absent π‘˜ z^{{*k}} is different from z k superscript 𝑧 π‘˜ z^{k} .

(In GAP, the image of a cyclotomic z 𝑧 z under * k absent π‘˜ *k can be computed as GaloisCyc( z , k 𝑧 π‘˜ z,k ).)

gap> ( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 );
-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
E(5)^2+E(5)^3

For Gcd ( n , k ) β‰  1 𝑛 π‘˜ 1 (n,k)\neq 1 , the map E ( n ) ↦ maps-to 𝑛 absent (n)\mapsto E ( n ) k superscript 𝑛 π‘˜ (n)^{k} does not define a field automorphism of ? n subscript ? 𝑛 ?_{n} but only a ? ? ? -linear map.

gap> GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 );
2
-6