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Thursday, November 21, 2013

Galois Traces and the Möbius Function (14.5.6,11)

Dummit and Foote Abstract Algebra, section 14.5, exercise 6:

MathJax TeX Test Page (6) Show TrQ(ζn)/Q(ζn)=μ(n) (11) Show the primitive nth roots of unity form a basis of Q(ζn) if and only if n is squarefree.

Proof: (6) Let n=pα11...pαkk. Since for any choice of primitives we have ζn=(ζpα11...ζpαkk)z for some z, by distributing powers we may simply assume ζn=ζpα11...ζpαkk. Since μ(ab)=μ(a)μ(b) for (a,b)=1, it suffices to show Tr(ζaζb)=Tr(ζa)Tr(ζb) for (a,b)=1 and that the proposition holds for n a prime power.

Note that for 1mn relatively prime to n, we have (ζpα11...ζpαkk)m=ζr1pα11...ζrkpαkk for 0ri<pαii and (ri,pαii)=1, and as well for such a selection of ri we have ζr1pα11...ζrkpαkk=(ζpα11...ζpαkk)m for some (m,n)=1 by the Chinese Remainder Theorem is the two-sided inverse of this operation. Since Tr(ζn)=(m,n)=1,1m<nζmn we therefore have Tr(ζpα11...ζpαkk)=Tr(ζpα11)...Tr(ζpαkk).

Now, we may observe Tr(ζp)=ζp+ζ2p+...+ζp1p=1. As well, assume q=pa for a2. Then Tr(ζq)=ζq+ζ2q+...+ζp1q+ζp+1q+...+ζ2p1q+ζ2p+1q+...+ζpa1q=(ζq+...+ζp1q)+ζpq(ζq+...+ζp1q)+ζ2pq(ζq+...+ζp1q)+...+ζp(pa11)q(ζq+...+ζp1q)=(1+ζpq+ζ2pq+...+ζp(pa11)q)(ζq+ζ2q+...+ζp1q)=f(ζpq)(ζq+ζ2q+...+ζp1q)where f(x)=xpa11x1 which is valid since ζpq1. Thus we note f(ζpq)=0 as (ζpq)pa11=ζpaq1=0 and so Tr(ζq)=0 and Tr(ζq) agrees with μ(q) on prime powers, and since both are relatively multiplicative the proposition is complete.

(11) () By the above, we see Tr(ζn)=0 if n is not squarefree. () Let n=p1p2...pk. Then we see Q(ζn)=Q(ζp1ζp2...ζpk) is the composite of Q(pk) and Q(ζp1...pk1). By induction on prime width, we see the latter field has for basis over Q the elements ζα1p1...ζαk1pk1 for 1αi<pi. Since the fields have relatively prime degree, we see the basis ζαkpk for 1αk<pk for Q(ζpk) over Q remains linearly independent over Q(ζp1...pk1) (Corollary 13.2.22). Hence a basis for Q(ζn) over Q is ζα1p1...ζαkpk for 1αi<pi, which by the Chinese Remainder Theorem are all the primitive nth roots. 

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