Finite dimensional factor algebras of
F
Miroslav Kureš
2[X1, …, Xn]
and their fixed point subalgebras
Institute of Mathematics
Brno University of Technology
Brno, Czech Republic
Abstract—Fixed point subalgebras of finite dimensional factor algebras of algebras of polynomials in n indeterminates over the
finite field
F
2(with respect to all
F
Keywords-polynomial; finite field; group of automorphisms; fixed point
2-algebra automorphisms) are fully described.
1. Introduction
In [6], we consider local commutative
R
-algebra Awith
identity, the nilpotent ideal nAof which has a finite dimension
as a vector space and A / nA=
R
and study its subalgebra SA of
fixed elements, SA ={aA;
M (a) = a for all M Aut
R
A}, where
Aut
R
Ais the group of
R
-automorphisms of the algebra A. This
research is motivated by differential geometry, wherealgebras
in question are usually called Weil algebras and, in particular,
the bijection between all natural operators lifting vector fields
from m-dimensional manifolds to bundles of Weil contact
elements and the subalgebra of fixed points SA of a Weil
algebra Awas determined (in [4]). Although in the known
geometrically motivated examples is usually SA =
R
(such SA
is called trivial), there are some algebras for which SA is a
proper superset of
R
In this paper, we simply replace
and they call attention to the geometry of
corresponding bundles. Thus, the fundamental problem is a
classification of algebras having SA nontrivial. See [4], [5] for
related geometric questions and the survey paper [8] for known
results up to now, especially for a number of claims concerning
the form of subalgebras of fixed points of various Weil
algebras.
R
by
F
2and study quite
analogous questions. We come to a different situation by this:
mainly, factor rings are finite
2. Polynomials over F2
rings (see [1]) and there is the
whole theory about this topic. It is known the ring
automorphism problem lying in a decision if a finite ring has a
non-identical automorphism or not. Results about fixed point
subalgebras are also qualitatively totally different from the real
case and they can have interesting applications in the coding
theory and cryptography.
Polynomials in nindeterminates over
F
a:
2are the maps of the
type
N
0no
F
i.e. a multiindex maps onto an element of
2,
F
2; the support of
the map must be by definition finite. We define the addition of
polynomials and the multiplication of polynomials by the
usual way and use also the standard denotation for them.
However, we consider only polynomials in this paper and not
their evaluations (polynomial maps); the fatal inaccuracy of
such a confusion is explained e.g. in [7]. With the mentioned
operations, polynomials over
F
2form the ring denoted
by
F
2[X1, …, Xn] or shortly by
F
II.1. Ideals in
F
2[X]
2[X].
The (unique) maximal ideal of
F
2[X] is
m = (X1, …, Xn).
Powers of m represent notable class of ideals.
We mention another important ideal. For finite fields
F
q, the
field ideal in
F
q[X1,…,Xn] isdefined as
f = (X1q –X1, …, Xn
q –Xn).
Thus, we have
f = (X12+X1, …, Xn
2+Xn).
for q= 2.
3. ALGEBRAS (
D
2)n
r
In this section, we will study factor rings
(
D
2)n
r
=
F
2[X1, …, Xn] / mr+1,
where r
N
.
III.1. Dual numbers over
F
2
As
D
=
R
[X] / (X2) is usually called the algebra of dual
numbers(which is definable promptly by
D
={a0+a1X; a0,
a1
R
, X2= 0}), we obtain forr = n = 1
D
2= (
D
2)11=
F
2[X] / (X2)
the algebra of dual numbers over
F
2. Elements of
D
2are ex-
pressible in the form
Published results were acquired using the subsidization of the GA
N
o. 201/09/0981.
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
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a0+a1X;a0, a1
F
2,X2= 0.
We observe that
D
2has the following additive and
multiplicative tables:
+ 0 1 X 1 +X
0
1
X
1 +X
0 1 X 1 +X
1 0 1 + X X
X 1 + X 0 1
1 + X X 1 0
× 0 1 X 1 +X
0
1
X
1 + X
0 0 0 0
0 1 X
1 + X
0 X 0 X
0 1 + X X 1
As to classification of B. Fine, [2], this finite ring can be
expressed as
(a,b; 2a= 2b= 0, a2= 0, b2=b, ab =a, ba =a).
(the case denoted by 'G' in [2]).
Furthermore, in [3] is presented that dual numbers over
F
2,
complex numbers over
F
2and paracomplex numbers over
F
2
are isomorphic rings.We find easily the following result.
(Analogously to the real case, by the subalgebra of fixed
elementsSA of an
F
2-algebra Awe mean the subalgebra of all
elements asatisfying
M (a) = a for all
F
2-automorphisms
M of A
and if SA=
F
2, we call SA trivial.)
PROPOSITION 1. The group of all
F
2-automorphisms of
D
2is
trivial. It follows that for A =
D
2is SA nontrivial as SA = A.
PROOF. Let M be an
F
2-automorphism of
D
2. As
M (1) = 1, M
is fully determined by a specification of
M (X). In general,
M (X) = b0+b1X;b0, b1
F
2.
However, we have
0 = M (0) = M (X2) = M (X) M (X) = b02+b12X2=b02;
thus, b0= 0, then, necessarily, b1= 1 for M be a bijection. So,
the group of all
F
2-automorphisms of
D
2contains only one
element: the identical automorphism. Then the rest of the
claim becomes evident. 
III.2. The case r>1, n=1
Elements of the algebra (
D
2)r= (
D
2)1r=
F
2[X] / mr+1 =
F
2[X]
/ (Xr+1 ) have a form
a0+a1X + a2X2+…+ arXr;a0, a1, a2, …, ar
F
2, Xr+1 = 0.
We start with the following lemma.
LEMMA 1. Every endomorphism M : (
D
2)ro(
D
2)rdeter-
mined by
M (1) = 1
M (X) = X+b2X2+…+brXr;b2, …, br
F
2
belongs to the group of all
F
2-automorphisms of (
D
2)r.
PROOF.It suffices to describe
M
: we have
Y =
M (X) = X+b2X2+ … + brX
Y2=X2+ terms of degree > 2
…
Yr-1 =X r-1+terms of degree > r-1
Yr=Xr
The last equation provides Xr by Y’s, the last but one provides
(after the substitution) Xr-1and so on. 
We recall that nAdenotes the ideal of nilpotent elements of
A(nilradical of A). If an element aAhas the property au = 0
for all unA, we call athe socle element of A. It is easy to find
that all socle elements constitute an ideal; this ideal is called the
socle of Aand denoted by soc(A). Now, we can formulate the
main result about automorphisms of (
D
2)rincluding also the
case r = 1.
PROPOSITION 2. For r
N
, let A= (
D
2)r.Every automor-
phism M : AoA has a form
M (1) = 1
M (X) = X+b2X2+ … + brXr;b2, …, br
F
2.
It follows SA is always nontrivial, in particular soc(A)SA.
PROOF. It is evident that the endomorphism
M (1) = 1
M (X) = b2X2+ … + brXr;b2, …, br
F
2
does not represent an automorphism. Further,
soc((
D
2)r )= {aXr;a
F
2}
and
M (Xr) = Xrwas demonstrated already in the previous lem-
ma. 
III.3. The case r=1, n>1
Elements of the algebra (
D
2 )n= (
D
2)n
1=
F
2[X1, …, Xn] / m2
=
F
2[X1, …, Xn] / (X1, …, Xn)2have a form
a0+a1X1+a2X2+…+anXn;a0, a1, a2, …, an
F
2,XiXj= 0 for
all i, j{1, …, n}.
PROPOSITION 3. The group of all
F
2-automorphisms of (
D
2)n
is isomorphic to the general linear group GL(n,2) = GL(n,
F
2)
of the order n over
F
2.
PROOF. A general form of endomorphisms of (
D
2)nis
M (1) = 1
M (X1) = b10+b11X1+b12X2+ … + b1nXn
M (X2) = b20+b21X1+b22X2+ … + b2nXn
…
M (Xn) = bn0+bn1X1+bn2X2+ … + bnnXn.
However, we have
0 = M (0) = M (X12) = M (X1) M (X1) = b102+b112X12+ … +
b1n
2Xn 2=b102,
thus, b10 = 0, and analogously, b20 = … = bn0= 0. Now, the
matrix (bij) must be invertible for
M be a bijection. So,
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automorphisms of (
D
2)ncorrespondsexactly with the group
GL(n,2). 
COROLLARY 1. Let i ,j^n}, izj.Then M
(i,j):(
D
2)no
(
D
2)ngiven by
M
(i,j)(1) = 1
M
(i,j)(Xi) = Xi+Xj
M
(i,j)(Xk) = Xkfor all k^ n}, kzi,
belongs to the group of all
F
2-automorphisms of (
D
2)n.
PROOF.It is clear that
M
(i,j) meets the form from the proof of
the previous proposition. 
REMARK 1.We remark that the order of GL(n, 2) is
3i=0n-1(2n-2i).
PROPOSITION 4. For n
N
, n > 1, let A= (
D
2)n.Then the
subalgebra SA of fixed points of A is always trivial.
PROOF.First, we prove that the element
X1+X2+… +Xn
is not fixed. For this, it suffices to take some automorphism
M
(i,j), e.g.
M
(1,2)sends X1+X2+ … + Xnonto X1+X3+ … + Xn.
Second, let {k1, …, kh}be a (non-empty) proper subset of
^n}, i.e. h < n. We provethat the element
Xk1+Xk2+ … + Xkh
is not fixed, too. We take i{k1, …, kh} and j^n}
{k1,…,kh} and apply M
(i,j): it sends Xk1+Xk2+ … + Xkhonto
Xk1+Xk2+ … + Xkh +Xj . So, SA =
F
2.
III.4. The case r>1, n>1
Elements of the algebra (
D
2)n
r=
F
2[X1, …, Xn] / mr+1=
F
2[X1, …, Xn] / (X1, …, Xn)r+1 have a form
a0+
a1X1 +a2X2 + … + anXn +
a11X12+a12 X1X2 + … + annXn
2+
…+
a1…1X1r+a1…12 X1r-1X2 + … + an…nXn
r,
a0,a1, …, an, a11, …, an…n 
F
2.
On basis of previous results we can find out nature of this
general case now.
PROPOSITION 5. For r
N
,n
N,
n>1,let A= (
D
2)n
r.Then
the subalgebra SA of fixed points of A is always trivial.
PROOF. Obviously, elements of GL(n,2) represent automor-
phisms also for (
D
2)nr.Of course, not all
Let PAand let exist i,j^ n} such that wPwXi zand
wPwXj Analogously with the case r= 1, n> 1, we apply
M
(i,j) for the demonstration that Pcannot be fixed.
automorphisms,
however, these (linear) automorphisms suffice for our
following considerations. In the proof, we use formally partial
derivations w wXjfor an expressing whether elements ofA
contain Xjin some non-zero power or not.
So, let QAis not of such a type and let Vbe a permutation
of n-tuple (X1, …, Xn) for which V(Q)zQ. As permutations of
(X1, …, Xn) are also elements of GL(n,2), we find again that Q
cannot be fixed.
Therefore we take RAsuch that wPwXizfor all
i^ n} and such that does not exist any permutation of
(X1, …, Xn) yielding a transformation of R. Nevertheless, a
"symmetry" of Rwill be again unbalanced by
M
(i,j), e.g.
M
(1,2).
Hence we have an automorphism for which not even Ris fixed.
Thus, only zero degreeelements of Aremain fixed with
respect to all automorphisms: SA is trivial. 
3. Summary
The previous assertions provide the following summary
theorem (r,n
N
).
THEOREM.The subalgebra SA of fixed points of A = (
D
2)n
r
is nontrivial if and only if n = 1. 
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polynomial automorphisms and the security of TTM
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