SIMPLE ALGEBRAS
C. BROWN AND S. PUMPL ¨UN
Abstract. We define nonassociative cyclic extensions of degreem of both fields and central simple algebras over fields. If a suitable field contains a primitivemth (resp.,qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degreem(resp.,qs). Some of Amitsur’s classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.
Introduction
Analogously as both for commutative field extensions [6, 2, 3, 26] and for associative
cyclic extensions of fields and central simple algebras [5], nonassocative cyclic extensions of degree m of a field or a central division algebra are investigated separately for prime
characteristics and for the case that the characteristic is zero or a primepwithgcd(p, m) = 1. Nonassociative cyclic extensions of degreepin characteristicpwere already studied in [18].
Let D be a finite-dimensional central division algebra over a field K. An (associative) central division algebra A over a field F is called a non-commutative cyclic extension of degreemofDoverK, if AutF(A) has a cyclic subgroup of automorphisms of ordermwhich
are all extended fromidD, and ifAis a free leftD-module of rankm[5]. For instance, ifF
contains a primitivemth root of unity, then generalized cyclic algebras (D, σ, a) are cyclic extensions of D of degree m [5, Theorem 6]. We recall that a generalized cyclic algebra
(D, σ, a) is a quotient algebraD[t;σ]/(tm−a)D[t;σ], whereD[t;σ] is a twisted polynomial ring,σ∈Aut(D) is an automorphism such thatσ|K has finite orderm, F0 = Fix(σ)∩K,
andf(t) =tm−a∈D[t;σ] with d∈F×
0 . The special case whereD =F andF0= Fix(σ) yields the cyclic algebra (F/F0, σ, a) [12, p. 19].
A finite-dimensional central simple algebra A overF is called aG-crossed product if it contains a maximal field extensionK/F which is Galois with Galois groupG= Gal(K/F).
IfGis solvable thenAis called asolvableG-crossed product. In [9] we revisited a result by Albert [1] on solvable crossed products and gave a proof for Albert’s result using generalized
cyclic algebras following Petit’s approach [17], proving that aG-crossed product is solvable if and only if it can be constructed as a chain of generalized cyclic algebras. Hence any
solvableG-crossed product division algebra is always a generalized cyclic division algebra.
In particular, hence if F contains a primitive mth root of unity, solvable crossed product division algebras overF are non-commutative cyclic extensions.
2010Mathematics Subject Classification. Primary: 17A35; Secondary: 17A60, 16S36.
Key words and phrases. Skew polynomial, Ore polynomial, cyclic algebra, cyclic extension.
A generalization of associative cyclic extensions of simple rings instead of division rings was considered in [13].
In this paper, we define and investigate nonassociative cyclic extensions of degree m of both fields and central simple algebras employing nonassociative generalized cyclic division
algebras: LetAbe a unital nonassociative division algebra. ThenAis called anonassociative cyclic extension ofD of degreem, ifAis a free leftD-module of rankmand Aut(A) has a cyclic subgroupGof orderm, such that for allH ∈G,H|D=idD.
We show that if F contains a primitive mth root of unity (i.e., F has characteristic 0 or characteristicpwith gcd(m, p) = 1), then the nonassociative generalized cyclic division
algebras (D, σ, a) =D[t;σ]/(tm−a)D[t;σ] witha∈D×are nonassociative cyclic extensions ofD of degreem. Additionally, the subgroup of orderm in AutF0(D, σ, a) that consists of
automorphisms extendingidD contains only inner automorphisms (Corollary 10). We also
investigate the structure of the automorphism groups of nonassociative generalized cyclic
algebras in general.
Note that nonassociative cyclic division algebras (K/F, σ, a) are a special case of
nonas-sociative generalized cyclic division algebras. IfF contains a primitivemth root of unity the nonassociative cyclic division algebras (K/F, σ, a) are nonassociative cyclic extensions ofK
of degreem. The subgroup of the automorphisms extendingidK has orderm, is isomorphic
to ker(NK/F), and contains only inner automorphisms. If F has no non-trivial mth root
of unity and a ∈ K× is not contained in any proper subfield of K, all automorphisms of (K/F, σ, a) are inner and leaveidK fixed (Theorem 3).
We point out that nonassociative generalized cyclic algebras have been recently success-fully used both in constructing space-time block codes and linear codes [19, 20, 21, 22].
The paper is organized as follows: After introducing the basic terminology in Section 1, we define nonassociative cyclic extensions and nonassociative generalized cyclic algebras in
Section 2 and investigate nonassociative cyclic extensions of a field. In Section 3 we show when generalized cyclic division algebras (D, σ, d) are nonassociative cyclic extensions ofD
of degreem. We briefly look at the question when a nonassociative overring is a nonasso-ciative cyclic extension of a field or a central simple algebra in Section 4.
The results presented in this paper complements the ones for the nonassociative algebras (K, δ, d) =K[t;δ]/K[t;δ]f(t) forf(t) =tp−t−d∈K[t;δ] constructed using a fieldK of
characteristicptogether with some derivationδwith minimum polynomialg(t) =tp−t∈
F[t], F = Const(δ), and of the nonassociative algebras (D, δ, d) = D[t;δ]/D[t;δ]f(t) for
f(t) =tp−t−d ∈D[t;δ] constructed using a division algebra D, where the derivation δ has minimum polynomialg(t) =tp−t∈F[t] and the field F characteristicp. [18].
1. Preliminaries
1.1. Nonassociative algebras. LetF be a field and letAbe anF-vector space. A is an
element inA, denoted by 1, such that 1x =x1 = xfor all x∈A. We will only consider unital algebras without saying so explicitly.
TheassociatorofAis given by [x, y, z] = (xy)z−x(yz). Theleft nucleusofAis defined as Nucl(A) ={x∈A|[x, A, A] = 0}, themiddle nucleusofAis Nucm(A) ={x∈A|[A, x, A] =
0}and theright nucleusofAis Nucr(A) ={x∈A|[A, A, x] = 0}. Nucl(A), Nucm(A), and
Nucr(A) are associative subalgebras ofA. Their intersection Nuc(A) ={x∈A|[x, A, A] =
[A, x, A] = [A, A, x] = 0} is the nucleus of A. Nuc(A) is an associative subalgebra of A containingF1 andx(yz) = (xy)z whenever one of the elementsx, y, z lies in Nuc(A). The
centerofAis C(A) ={x∈A|x∈Nuc(A) andxy=yxfor ally∈A}.
An algebraA6= 0 is called adivision algebraif for anya∈A,a6= 0, the left multiplication witha,La(x) =ax, and the right multiplication witha,Ra(x) =xa, are bijective. IfAhas
finite dimension overF,Ais a division algebra if and only ifAhas no zero divisors [24, pp. 15, 16]. An element 06=a∈A has aleft inverse al∈A, ifRa(al) =ala= 1, and aright inverse ar∈A, ifLa(ar) =aar= 1. Ifmr=ml then we denote this element bym−1.
An automorphism G∈AutF(A) is aninner automorphism if there is an elementm∈A
with left inverseml such that G(x) = (mlx)m for allx∈A. We denote such an
automor-phism byGm. The set of inner automorphisms{Gm|m∈Nuc(A) invertible}is a subgroup
of AutF(A). Note that if the nucleus of A is commutative, then for all 06=n ∈ Nuc(A),
Gn(x) = (n−1x)n=n−1xnis an inner automorphism ofAsuch thatGn|Nuc(A)=idNuc(A).
1.2. Division algebras obtained from twisted polynomial rings. Let D be a unital division ring andσa ring automorphism ofD. Thetwisted polynomial ring D[t;σ] is the set of polynomialsa0+a1t+· · ·+antn with ai∈D, where addition is defined term-wise and
multiplication byta=σ(a)t (a∈D) [16]. That means, atnbtm=aσn(b)tn+mandtna=
σn(a)tn for all a, b∈D [12, p. 2]. R=D[t;σ] is a left principal ideal domain and there is
a right division algorithm inR, i.e. for allg, f∈R,f 6= 0, there exist uniquer, q∈Rsuch that deg(r)<deg(f) andg =qf+r[12, p. 3]. (Our terminology is the one used by Petit
[17] and different from Jacobson’s [12], who calls what we call right a left division algorithm and vice versa.)
An elementf ∈Risirreducible inRif it is no unit and it has no proper factors, i.e there do not existg, h∈R such thatf =gh[12, p. 11].
Let f ∈D[t;σ] be of degreemand let modrf denote the remainder of right division by
f. Then the vector space Rm={g∈D[t;σ]|deg(g)< m} together with the multiplication
g◦h=ghmodrf
becomes a unital nonassociative algebraSf = (Rm,◦) overF0={z∈D|zh=hzfor allh∈ Sf} (cf. [17, (7)]), andF0is a subfield of D. We also denote this algebraR/Rf.
We note that when deg(g) + deg(h)< m, the multiplication ofgandhinSf is the same
as the multiplication ofgandhinR [17, (10)].
A twisted polynomial f ∈R is right-invariant if f R⊂Rf. Iff is right invariant then Rf is a two-sided ideal and conversely, every two-sided ideal inR arises this way.
Sf is associative if and only iff is right-invariant. In that case,Sf is the usual quotient
2. Nonassociative generalized cyclic algebras and nonassociative cyclic
extensions
In the following, let D be a division algebra which is finite-dimensional over its center F= C(D) andσ∈Aut(D) such thatσ|F has finite ordermand fixed fieldF0= Fix(σ)∩F.
Note thatF/F0is automatically a cyclic Galois field extension of degreemwith Gal(F/F0) =
hσ|Fi.
2.1. Following Jacobson [12, p. 19], an (associative)generalized cyclic algebra is an associa-tive algebraSf =D[t;σ]/D[t;σ]f constructed using a right-invariant twisted polynomial
f(t) =tm−d∈D[t;σ]
withd∈F0×. We write (D, σ, d) for this algebra. IfD is a central simple algebra overF of degreen, then (D, σ, d) is a central simple algebra overF0of degreemnand the centralizer
ofD in (D, σ, d) isF [12, p. 20]. In particular, ifD=F,F/F0 is a cyclic Galois extension of degreem with Galois group generated byσ and f(t) =tm−d∈F[t;σ], we obtain the
cyclic algebra (F/F0, σ, d) of degreem.
This definition generalizes to nonassociative algebras as follows:
Definition 1. Anonassociative generalized cyclic algebra of degreemn is an algebraSf =
D[t;σ]/D[t;σ]f overF0 with f(t) =tm−d∈D[t;σ], d∈D×. We denote this algebra by
(D, σ, d).
The algebraA= (D, σ, d),d∈D×, has dimensionm2n2 overF0. In particular, ifD=F andF/F0 is a cyclic Galois extension of degree mwith Galois group generated byσ, then
(F/F0, σ, d) is a nonassociative cyclic algebra [25]. Ais associative if and only if d∈F0. If (D, σ, d) is not associative then Nucl(A) = Nucm(A) = D and Nucr(A) = {g ∈ Sf|f g ∈
Rf}.
(D, σ, d) is a division algebra overF0 if and only iff(t) =tm−d∈D[t;σ] is irreducible
[17, (7)]. Moreover, we know thatf(t) =t2−d∈D[t;σ] is irreducible if and only ifσ(z)z6=d for allz∈D,f(t) =t3−d∈D[t;σ] is irreducible if and only ifd6=σ2(z)σ(z)zfor allz∈D,
andf(t) =t4−d∈D[t;σ] is irreducible if and only if
σ2(y)σ(y)y+σ2(x)y+σ2(y)σ(x)6= 0 orσ2(x)x+σ2(y)σ(y)x6=d
for all x, y ∈ D (cf. [17] or [19], [7, Theorem 3.19], see also [11]). More generally, if F0 contains a primitive mth root of unity and m is prime then f(t) = tm−d ∈ D[t;σ] is
irreducible if and only ifd6=σm−1(z)· · ·σ(z)zfor allz∈D ([7, Theorem 3.11], see also [19,
Theorem 6]).
Amitsur’s definition [5] of cyclic extensions generalizes to the nonassociative setting as
follows:
Definition 2. Letm≥2. LetAbe a nonassociative division algebra with centerF0andD an associative division algebra with centerF. ThenA is a nonassociative cyclic extension
2.2. Nonassociative cyclic extensions of a field. For a nonassociative cyclic algebra (K/F, σ, d) of degreem, and for allk∈K such thatNK/F(k) = 1, the map
Hid,k( m−1
X
i=0
aiti) =a0+
m−1
X
i=1 ai
i−1
Y
l=0 σl(k)
ti
is an innerF-automorphism of (K/F, σ, d) extendingidK. The subgroup generated by the
automorphismsHid,k is isomorphic to ker(NK/F) [8, Theorem 19].
The maps Hid,k are the only F-automorphisms of (K/F, σ, d), unless for some j ∈ {1, . . . , m−1}, σj can be extended to an F-automorphism of (K/F, σ, d) as well. More
precisely, the automorphism τ = σj with j ∈ {1, . . . , m−1} can be extended to an F
-automorphismH of (K/F, σ, d), if and only if there is an elementk∈Ksuch that
(1) σj(d) =NK/F(k)d.
The extension then has the formH =Hτ,k with
(2) Hτ,k(
m−1
X
i=0
aiti) =τ(a0) +
m−1
X
i=1 τ(ai)
i−1
Y
l=0 σl(k)
ti
[8, Theorem 4]. We then immediately get the following partial generalization of [5, Theorem
6]:
Theorem 1. Suppose F contains a primitive mth root of unity ω, A = (K/F, σ, d) is a nonassociative cyclic division algebra of degree m over F, and d ∈ K\F. Then A is a nonassociative cyclic extension of K of degree m. The generating automorphism of the subgroup ofAutF(A)of ordermis given by Hid,ω.
Proof. hHid,ωiis a cyclic subgroup of AutF(A) of ordermby [8, Theorem 20]. It consists of
automorphisms extendingidK, thereforeAis a nonassociative cyclic extension of K.
Corollary 2. Ifm is prime,F contains a primitivemth root of unity andK/F is a cyclic Galois extension of degreem, thenK has a nonassociative cyclic extension of degreem.
Proof. Let d∈K\F and suppose Gal(K/F) =hσi. Then sincem is prime, the nonasso-ciative cyclic algebraA= (K/F, σ, d) is a division algebra [25, Corollary 4.5]. ThusA is a
nonassociative cyclic extension ofK by Theorem 1.
IfF has no non-trivialmth root of unity, we obtain:
Theorem 3. Suppose F has no non-trivial mth root of unity. Let A = (K/F, σ, d) be a nonassociative cyclic algebra of degree m where d ∈ K× is not contained in any proper subfield ofK. Then everyF-automorphism ofA leaves K fixed and
AutF(A)∼= ker(NK/F).
Proof. Every automorphism ofAhas the formHid,k: suppose that there existj∈ {1, . . . , m−
1}andk∈K× such thatH
σj,k∈AutF(A). This impliesH2
σj,k=Hσj,k◦Hσj,k∈AutF(A) and
Hσ2j,k
mX−1
i=0 xiti
=σ2j(x0) +
m−1
X
i=1
σ2j(xi)
iY−1
q=0
σj+q(k)σq(k)ti. (3)
NowH2
σj,k must have the form Hσ2j,l for some l∈ K×, and comparing (2) and (3) yields l = kσj(k). Similarly, Hσ3j,k = Hσ3j,s ∈ AutF(A) where s = kσj(k)σ2j(k). Continuing
in this manner we conclude that the automorphismsHσj,k, Hσ2j,l, Hσ3j,s, . . .all satisfy (1)
implying that
σj(d) =NK/F(k)d,
σ2j(d) =NK/F(kσj(k))d=NK/F(k)2d,
..
. ...
d=σnj(d) =NK/F(k)nd,
(4)
wheren=m/gcd(j, m) is the order ofσj. Note thatσij(d)6=dfor alli∈ {1, . . . , n−1}since
dis not contained in any proper subfield ofK. ThereforeNK/F(k)n= 1 andNK/F(k)i 6= 1
for alli∈ {1, . . . , n−1} by (4), i.e. NK/F(k) is a primitive nth root of unity, thus also an
mth root of unity, a contradiction. This proves the assertion.
Note that ifd∈K× is not contained in any proper subfield ofKthen 1, d, . . . , dm−1 are
linearly independent overF and thusAis a division algebra [25]. In particular, ifmis prime then 1, d, . . . , dm−1 are linearly independent overF. This yields for a field F of arbitrary
characteristic:
Corollary 4. Suppose that F has no non-trivial mth root of unity. If d ∈ K× is not contained in any proper subfield of K (e.g. if m is prime), and ker(NK/F)has a subgroup of orderm, then any cyclic algebraA= (K/F, σ, d)is a cyclic extension of K of degreem.
Example 5. LetK=Fqm be a finite field,q=pr for some primep,σan automorphism of K of orderm≥2 andF = Fix(σ) =Fq,i.e. K/F is a cyclic Galois extension of degreem
with Gal(K/F) =hσi.Then ker(NK/F) is a cyclic group of orders= (qm−1)/(q−1) and
any division algebra (K/F, σ, d) has exactly sinner automorphisms, all of them extending idK. The subgroup they generate is cyclic and isomorphic to ker(NK/F) [10]. Hence if
m divides s, which is the case if F contains a primitive mth root of unity, then there is a subgroup of automorphisms of order m extendingidK and hence (K/F, σ, d) is a cyclic
extension ofKof degreem.
3. Nonassociative cyclic extensions of a central simple algebra
Theorem 6. (i) Suppose τ ∈ AutF0(D) commutes with σ. Then τ can be extended to an automorphism H ∈ AutF0(A), if and only if there is some k ∈ F× such that τ(d) =
NF /F0(k)d. In that case, the extensionH of τ has the form H =Hτ,k with
Hτ,k( m−1
X
i=0
aiti) =τ(a0) +
m−1
X
i=1 τ(ai)
i−1
Y
l=0 σl(k)
ti.
All mapsHτ,k where τ ∈AutF0(D) commutes withσ and where k ∈F
× such that τ(d) =
NF /F0(k)d(hence NF /F0(k)
mn= 1), are automorphisms ofA.
In particular, for τ6=idandd6∈Fix(τ),NF /F0(k)6= 1.
(ii) id∈Aut(D) can be extended to an automorphism H ∈AutF0(A), if and only if there is some k∈F× such that NF /F0(k) = 1. In that case, the extension H of id has the form
H=Hid,k with
Hid,k( m−1
X
i=0
aiti) =a0+
m−1
X
i=1 ai
i−1
Y
l=0 σl(k)
ti.
AllHid,k wherek∈F× such thatNF /F0(k) = 1 are automorphisms ofA.
Proof. (i) Let H ∈AutF0(A), thenH|D∈AutF0(D), sinceH leaves the left nucleus
invari-ant. ThusH|D=τ for someτ∈AutF0(D). WriteH(t) =P m−1
i=0 kitifor someki∈D, then
we have
H(tz) =H(t)H(z) =
m−1
X
i=0 kiti
τ(z) =
m−1
X
i=0
kiσi(τ(z))ti,
and
H(tz) =H(σ(z)t) =τ(σ(z))
m−1
X
i=0 kiti=
m−1
X
i=0
τ(σ(z))kiti
for allz∈D. Comparing the coefficients ofti yields
kiσi(τ(z)) =kiτ(σi(z)) =τ(σ(z))ki for alli={0, . . . , m−1}
for allz∈D sinceσand τ commute. In particular, we obtain
ki(τ(σi(z))−τ(σ(z))) = 0 for alli∈ {0, . . . , m−1}
for allz∈F, i.e. ki= 0 orσ|F =σi|F for alli∈ {0, . . . , m−1}. As σ|F has orderm, this
meanski= 0 for all 16=i∈ {0, . . . , m−1}. Fori= 1, this yieldsk1τ(σ(z)) =τ(σ(z))k1 for allz∈D, hencek1∈F. This impliesH(t) =ktfor somek∈F×.
Since
H(zti) =H(z)H(t)i=τ(z)(kt)i=τ(z)
i−1
Y
l=0
σl(k)ti,
for alli∈ {1, . . . , m−1}and allz∈D, H has the form
Hτ,k: m−1
X
i=0
aiti7→τ(a0) +
m−1
X
i=1 τ(ai)
i−1
Y
l=0 σl(k)
ti,
for somek∈F×.
Comparing the constant terms in H(t)m=H(tm) =H(d) implies
LetN =NF /F0◦ND/F be the norm of theF0-algebraD. ApplyingN to both sides of the equation yieldsN(d) =N(k)mN(d), so thatN(k)m= 1. Nowk∈F× andD has degreen,
thus
N(k) =NF /F0(ND/F(k)) =NF /F0(k n) =N
F /F0(k) n,
and soN(k)m=NF /F0(k) nm= 1.
Finally, the fact that the maps Hτ,k are automorphisms whenτ commutes with σ, and
τ(d) =NF /F0(k)d,can be shown similarly to the proof of [8, Theorem 4], see also [7].
(ii) In particular, forτ=id, we get from (i) thatH has the form
Hid,k: m−1
X
i=0
aiti7→a0+
m−1
X
i=1 ai
i−1
Y
l=0 σl(k)
ti
for somek∈F× withkσ(k)· · ·σm−1(k) =NF /F0(k) = 1.
The above is proved for a more general set-up in the first author’s PhD thesis [7]. Note
that the automorphismsHτ,k are restrictions of automorphisms of the twisted polynomial
ringD[t;σ].
Corollary 7. (i) The subgroup of F0-automorphisms of A extending idD ∈ AutF0(D) is isomorphic to
{k∈F×|kσ(k)· · ·σm−1(k) = 1}.
(ii) If F0 contains a primitive mth root of unity ω, then hHid,ωi is a cyclic subgroup of AutF0(A)of orderm.
3.2. We obtain the following generalization of [5, Theorem 6]:
Corollary 8. SupposeF0 contains a primitivemth root of unity. Iff(t) =tm−d∈D[t;σ]
is irreducible, thenA is a nonassociative cyclic extension of D of degreem. In particular, ifmis prime and
d6=σm−1(z)· · ·σ(z)z
for allz∈D, then Ais a nonassociative cyclic extension ofD of degreem.
Proof. If F0 contains a primitive mth root of unity ω, then hHid,ωi is a cyclic subgroup of AutF0(A) of order m by Corollary 7 (ii). If m is prime, then f(t) = tm−d ∈ D[t;σ] is
irreducible if and only if
d6=σm−1(z)· · ·σ(z)z
for allz∈D. The rest is trivial.
Proposition 9. Every automorphism Hid,k of Ais an inner automorphism
Gc( m−1
X
i=0
aiti) = (c−1 m−1
X
i=0 aiti)c
Proof. For all k∈F such thatNF /F0(k) = 1,Hid,k is an F-automorphism extendingidD.
These are the onlyF0-automorphisms ofA, unlessτ6=idcan be also extended. By Hilbert’s
Satz 90,NF /F0(k) = 1 if and only if there isc∈F× such thatk=c−1σ(c) [14]. So there is
c∈F× such thatk=c−1σ(c) and
kσ(k)· · ·σi−1(k) =cσi(c), i= 1. . . . , m−1
yields thatHid,k =Gwith
G(
m−1
X
i=0
aiti) =a0+a1c−1σ(c)t+
m−1
X
i=2
aic−1σi(c)ti,
which is an inner automorphism, sinceG=Gc with
Gc( m−1
X
i=0
aiti) = (c−1 m−1
X
i=0 aiti)c.
Note that here we use thatF=C(D).
Corollary 10. If F0 contains a primitive mth root of unityω, thenA is a cyclic extension
ofD of orderm, and all automorphisms extending idD are inner.
Example 11. LetF andLbe fields and letKbe a cyclic Galois extension of bothF andL such that [K:F] =n, [K:L] =m, Gal(K/F) =hγiand Gal(K/L) =hσi, andσ◦γ=γ◦σ. DefineF0=F∩L.
LetD= (K/F, γ, c) be a cyclic division algebra of degreenwithc∈F0, i.e. D∼=D0⊗F0K
for some cyclic algebra D0 = (F/F0, γ, c). Let 1, e. . . . , en−1 be the canonical basis of D,
that isen=c, ex=γ(x)efor everyx∈K. Forx=x
0+x1e+x2e2+· · ·+xn−1en−1∈D, define anL-linear mapσ∈AutL(D) via
σ(x) =σ(x0) +σ(x1)e+σ(x2)e2+· · ·+σ(xn−1)en−1
(note thatc∈L impliesσ(xy) =σ(x)σ(y) for allx, y∈D). Thenσ∈AutF0(D) has order
m. For alld∈D×,
D[t;σ]/D[t;σ](tm−d) = (D, σ, d)
is a generalized nonassociative cyclic algebra of degree mn over F0 (used for instance in [20]). (D, σ, d) is associative if and only ifd∈F0. In the special case thatd∈F×,
(D, σ, d) = (L/F0, γ, c)⊗F0(F/F0, σ, d)
is the tensor product of an associative and a nonassociative cyclic algebra.
IfF0 contains a primitivemth root of unity andd∈D×\F0 is chosen such thatf(t) = tm−d ∈ D[t;σ] is irreducible, then (D, σ, d) is a cyclic extension of D of order m, and
all automorphisms extendingidD are inner (Corollary 10). Recall that ifm is prime then
f(t) =tm−d∈D[t;σ] is irreducible if and only if d6=σm−1(z)· · ·σ(z)z for allz∈D.
3.3. In the following, letDbe a division algebra which is finite-dimensional over its center F = C(D), σ ∈ Aut(D) an automorphism such that σ|F has finite order q and fixed
field F0 = Fix(σ)∩F. If D has degree n then the associative generalized cyclic algebra A= (D, σ, a) has degreeqnoverF0. We choosea∈F0 such thatAis a division algebra.
Now assume F0 contains a primitive qth root of unity ω. Then τ = HidD,ω : A → A
generates a cyclic subgroup of AutF0(A) of order q by [5, Theorem 6] which consists of
automorphisms which all extendidD. We obtain the following generalization of [5, Theorem
7]:
Theorem 12. Suppose there existsρ∈Aut(A),b∈Aand16=k∈F0 such that
(1)τ commutes withρ, (2)τ(b) =kρ(k)· · ·ρm−1(k)b,
(3)kq is a primitivemth root of unity, (4)tm−b∈A[t;ρ]is irreducible, and
(5) the algebra B =A[t;ρ]/A[t;ρ](tm−b) is either associative, or finite-dimensional over
F0∩Fix(ρ), or finite-dimensional overNucr(B).
ThenB is a nonassociative cyclic extension ofD of degreemq which containsA.
Proof. Since B is a free leftA-module of rank m andA is a free left D-module of rankq, B is a free left D-module of rankmq. Furthermore, (4) and (5) yield thatB is a division algebra by [17, (7)]. Define the map
Hτ,k:B→B, m−1
X
i=0
xiti7→τ(x0) +
m−1
X
i=1 τ(xi)
i−1
Y
l=0 ρl(k)
ti (xi∈A),
then (1) and (2) together imply thatHτ,k is an automorphism ofB by [8, Theorem 4].
Hτ,k has order mq: We haveτ(k) =k because k∈F0 ⊂D. Therefore straightforward
calculations yield H2
τ,k =Hτ,k◦Hτ,k =Hτ2,kτ(k) =Hτ2,k2, Hτ,k3 =Hτ3,k3 etc., thus Hτ,k
will have order at leastq. Afterq steps we obtainHτ,kq =HidA,v with v=k
q and soH τ,k
has ordermq by (3).
FinallyHτ,k|D=τ|D=idD, hence we concludeB is a nonassociative cyclic extension of
Dof degreemq.
4. When is a ring a nonassociative cyclic extension?
A nonassociative ringA6= 0 is called aright division ring, if for alla∈A,a6= 0, the right multiplication witha,Ra(x) =xa, is bijective. IfD is a division ring andf is irreducible,
thenSf =D[t;σ]/D[t;σ]f is a right division algebra and has no zero divisors ([17, (6)] or
[11]).
Theorem 13. (cf. [17, (3), (6)])
(i) LetS be a nonassociative ring with multiplication◦. Suppose that
(1)S has an associative subringD which is a division algebra andS is a free leftD-module of rankm, and there ist∈S such that tj,0≤i < mis a basis ofS overD, when defining
tj+1=t◦tj,t0= 1;
(3) for alla, b, c∈D,i+j < m,k < m, we have[a◦ti, b◦tj, c◦tk] = 0.
ThenS ∼=Sf with f(t) ∈D[t;σ, δ] and σ, δ defined via t◦a =σ(a)◦t+δ(a) and where the polynomialf(t) =tm−Pm−1
i=0 dit
i is given bytm=Pm−1
i=0 dit
i witht0= 1, ti+1=t◦ti,
0≤i < m.
(ii) IfS is a right division ring in (i) then f is irreducible.
Theorem 13 yields the nonassociative analogues to the existence conditions for associative cyclic extensions in [5, Theorem 6].
Theorem 14. (i) Let S be a nonassociative ring with multiplication◦, which has a field K
as a subring, and is a free leftK-vector space of dimensionm. Suppose that
(1) there is t ∈ S such that ti, 0 ≤i < m, is a basis of S over K when defining t0 = 1, ti+1=t◦ti,0≤i < m;
(2) for alla∈K,a6= 0, there isa0 ∈K×, such thatt◦a=a0◦t; (3) for alla, b, c∈K,i+j < m,k < m, we have[a◦ti, b◦tj, c◦tk] = 0; (4)tm=d for somed∈K×;
(5) the mapσ:K→K, σ(a) =a0, has order m and fixed fieldF ={a∈K|t◦a=a◦t} containing a primitivemth root of unityω, andK/F is a finite cyclic Galois extension. ThenS ∼=Sf= (K/F, σ, d)withf(t) =tm−d∈K[t;σ].
(ii) IfS is a right division ring in (i) thenf is irreducible andS ∼= (K/F, σ, d)is a nonas-sociative cyclic extension ofK of degreem.
Proof. (1), (2) and (3) imply thatS∼=Sf withf ∈K[t;σ] andσdefined viat◦a=σ(a)◦t,
i.e. σ(a) =a0,and where the polynomialf(t) =tm−Pm−1
i=0 ditiis given bytm=P m−1
i=0 diti
for some suitably chosen di (cf. [17, (3)]). (4) implies that indeed f(t) = tm−d. (5)
guarantees that (K/F, σ, d) whereF contains a primitivemth root of unityω.
(ii) Here we are in the setup of Theorem 1 which yields the assertion:F contains a primitive
mth root of unity ω, so hHid,ωi is a cyclic subgroup of order m of the division algebra
(K/F, σ, d).
For nonassociative cyclic extensions of a central simple algebraDwe obtain from Theorem 13:
Theorem 15. (i) Let S be a nonassociative ring with multiplication ◦, which has an as-sociative subring D which is a division algebra and S is a free left D-module of rank m. Suppose that
(1) there is t ∈ S such that ti, 0 ≤ i < m, is a basis of S over D when defining t0 = 1, ti+1=t◦ti,0≤i < m;
(2) for alla∈D,a6= 0, there are a0∈D,a06= 0, such that t◦a=a0◦t; (3) for alla, b, c∈D,i+j < m,k < m, we have[a◦ti, b◦tj, c◦tk] = 0; (4)tm=d;
(5) the mapσ:D→D,σ(a) =a0, has orderm, fixed field {a∈D|t◦a=a◦t} andD/F
is a central simple algebra, whereF0=F∩Fix(σ)contains a primitivemth root of unityω.
ThenS ∼=Sf= (D, σ, d)with f(t) =tm−d∈D[t;σ].
Proof. (1), (2) and (3) imply thatS∼=Sf withf ∈D[t;σ] andσdefined viat◦a=σ(a)◦t,
i.e. σ(a) =a0,and where the polynomialf(t) =tm−Pm−1
i=0 ditiis given bytm=P m−1
i=0 diti
for some suitably chosendi (cf. [17, (3)]). (4) implies f(t) =tm−d. (5) guarantees that
S∼= (D, σ, d) whereF contains a primitivemth root of unityω.
(ii) Here we are in the setup of Theorem 6 which yields the assertion, sinceF contains a
primitivemth root of unityω,hHid,ωiis a cyclic subgroup of ordermof the division algebra
(D, σ, d).
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