60.3-1 ZumbroichBase
β€£ ZumbroichBase( n, m )( function )

Let n and m be positive integers, such that m divides n. ZumbroichBase returns the set of exponents i 𝑖 i for which E(n)^ i 𝑖 i belongs to the (generalized) Zumbroich basis of the cyclotomic field ? n subscript ? 𝑛 ?_{n} , viewed as a vector space over ? m subscript ? π‘š ?_{m} .

This basis is defined as follows. Let P 𝑃 P denote the set of prime divisors of n, 𝑛 = ∏ p ∈ P p Ξ½ p n subscript product 𝑝 𝑃 superscript 𝑝 subscript 𝜈 𝑝 \textit{n}=\prod_{{p\in P}}p^{{\nu_{p}}} , and π‘š = ∏ p ∈ P p ΞΌ p m subscript product 𝑝 𝑃 superscript 𝑝 subscript πœ‡ 𝑝 \textit{m}=\prod_{{p\in P}}p^{{\mu_{p}}} with ΞΌ p ≀ Ξ½ p subscript πœ‡ 𝑝 subscript 𝜈 𝑝 \mu_{p}\leq\nu_{p} . Let e l = subscript 𝑒 𝑙 absent e_{l}= E ( l ) 𝑙 (l) for any positive integer l 𝑙 l , and { e n 1 j } j ∈ J βŠ— { e n 2 k } k ∈ K = { e n 1 j β‹… e n 2 k } j ∈ J , k ∈ K tensor-product subscript superscript subscript 𝑒 subscript 𝑛 1 𝑗 𝑗 𝐽 subscript superscript subscript 𝑒 subscript 𝑛 2 π‘˜ π‘˜ 𝐾 subscript β‹… superscript subscript 𝑒 subscript 𝑛 1 𝑗 superscript subscript 𝑒 subscript 𝑛 2 π‘˜ formulae-sequence 𝑗 𝐽 π‘˜ 𝐾 \{e_{{n_{1}}}^{j}\}_{{j\in J}}\otimes\{e_{{n_{2}}}^{k}\}_{{k\in K}}=\{e_{{n_{1% }}}^{j}\cdot e_{{n_{2}}}^{k}\}_{{j\in J,k\in K}} .

Then the basis is

\[ B_{{n,m}} = \bigotimes_{{p \in P}} \bigotimes_{{k = \mu_p}}^{{\nu_p-1}} \{ e_{{p^{{k+1}}}}^j \}_{{j \in J_{{k,p}}}} \]

where J k , p = subscript 𝐽 π‘˜ 𝑝 absent J_{{k,p}}=

{ 0 } 0 \{0\} ; k = 0 , p = 2 formulae-sequence π‘˜ 0 𝑝 2 k=0,p=2
{ 0 , 1 } 0 1 \{0,1\} ; k ⁒ & ⁒ g ⁒ t ; 0 , p = 2 π‘˜ & 𝑔 𝑑 0 𝑝 2 k>0,p=2
{ 1 , … , p - 1 } 1 … 𝑝 1 \{1,\ldots,p-1\} ; k = 0 , p β‰  2 formulae-sequence π‘˜ 0 𝑝 2 k=0,p\neq 2
{ - ( p - 1 ) / 2 , … , ( p - 1 ) / 2 } 𝑝 1 2 … 𝑝 1 2 \{-(p-1)/2,\ldots,(p-1)/2\} ; k ⁒ & ⁒ g ⁒ t ; 0 , p β‰  2 π‘˜ & 𝑔 𝑑 0 𝑝 2 k>0,p\neq 2

Β 


B n , 1 subscript 𝐡 𝑛 1 B_{{n,1}} is equal to the basis of ? n subscript ? 𝑛 ?_{n} over the rationals which is introduced inΒ [Zum89]. Also the conversion of arbitrary sums of roots of unity into its basis representation, and the reduction to the minimal cyclotomic field are described in this thesis. (Note that the notation here is slightly different from that there.)

B n , m subscript 𝐡 𝑛 π‘š B_{{n,m}} consists of roots of unity, it is an integral basis (that is, exactly the integral elements in ? n subscript ? 𝑛 ?_{n} have integral coefficients w.r.t.Β  B n , m subscript 𝐡 𝑛 π‘š B_{{n,m}} , cf.Β IsIntegralCyclotomic (18.1-4)), it is a normal basis for squarefree n 𝑛 n and closed under complex conjugation for odd n 𝑛 n .

Note: For 𝑛 ≑ 2 ( mod 4 ) n annotated 2 pmod 4 \textit{n}\equiv 2\pmod{4} , we have ZumbroichBase(n, 1) = 2 * ZumbroichBase(n/2, 1) and List( ZumbroichBase(n, 1), x -> E(n)^x ) = List( ZumbroichBase(n/2, 1), x -> E(n/2)^x ).

gap> ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );
[ 1, 2, 4, 7, 8, 11, 13, 14 ]
[ 0, 3 ]
gap> ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );
[ 2, 4, 6, 8 ]
[ 0, 1, 2, 3, 4, 5, 6, 7 ]