On maximal and minimal linear matching property
The matching basis in field extentions is introduced by S. Eliahou and C. Lecouvey in [2]. In this paper we define the minimal and maximal linear matching property for field extensions and prove that if K is not algebraically closed, then K has minimal linear matching property. In this paper we will...
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irk-123456789-1523002019-06-10T01:25:58Z On maximal and minimal linear matching property Aliabadi, M. Darafsheh, M.R. The matching basis in field extentions is introduced by S. Eliahou and C. Lecouvey in [2]. In this paper we define the minimal and maximal linear matching property for field extensions and prove that if K is not algebraically closed, then K has minimal linear matching property. In this paper we will prove that algebraic number fields have maximal linear matching property. We also give a shorter proof of a result established in [6] on the fundamental theorem of algebra. 2013 Article On maximal and minimal linear matching property / M. Aliabadi, M.R. Darafsheh // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 2. — С. 174–178. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC:12F05. http://dspace.nbuv.gov.ua/handle/123456789/152300 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The matching basis in field extentions is introduced by S. Eliahou and C. Lecouvey in [2]. In this paper we define the minimal and maximal linear matching property for field extensions and prove that if K is not algebraically closed, then K has minimal linear matching property. In this paper we will prove that algebraic number fields have maximal linear matching property. We also give a shorter proof of a result established in [6] on the fundamental theorem of algebra. |
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Aliabadi, M. Darafsheh, M.R. |
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Aliabadi, M. Darafsheh, M.R. On maximal and minimal linear matching property Algebra and Discrete Mathematics |
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Aliabadi, M. Darafsheh, M.R. |
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Aliabadi, M. |
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On maximal and minimal linear matching property |
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On maximal and minimal linear matching property |
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On maximal and minimal linear matching property |
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On maximal and minimal linear matching property |
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On maximal and minimal linear matching property |
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on maximal and minimal linear matching property |
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Інститут прикладної математики і механіки НАН України |
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On maximal and minimal linear matching property / M. Aliabadi, M.R. Darafsheh // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 2. — С. 174–178. — Бібліогр.: 7 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT aliabadim onmaximalandminimallinearmatchingproperty AT darafshehmr onmaximalandminimallinearmatchingproperty |
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2025-07-13T02:46:46Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 15 (2013). Number 2. pp. 174 – 178
c© Journal “Algebra and Discrete Mathematics”
On maximal and minimal
linear matching property
M. Aliabadi, M. R. Darafsheh
Communicated by V. Dlab
Abstract. The matching basis in field extentions is in-
troduced by S. Eliahou and C. Lecouvey in [2]. In this paper we
define the minimal and maximal linear matching property for field
extensions and prove that if K is not algebraically closed, then K
has minimal linear matching property. In this paper we will prove
that algebraic number fields have maximal linear matching property.
We also give a shorter proof of a result established in [6] on the
fundamental theorem of algebra.
1. Introduction
Throughout this paper we will consider a field extension K ⊂ L where
K is commutative and central in L. Let G be an additive group and
A, B ⊂ G be nonempty finite subsets of G. A matching from A to B is a
map φ : A → B which is bijective and satisfies the condition
a + φ(a) 6∈ A
for all a ∈ A. This notion was introduced in [3] by Fan and Losonczy,
who used matchings in Z
n as a tool for studying an old problem of
Wakeford concerning canonical forms for symmetric tensors [7]. Eliahou
2010 MSC: 12F05.
Key words and phrases: Linear matching property, Algebraic number field,
Field extension, Maximal linear matching property, Minimal linear matching property.
M. Aliabadi, M. R. Darafsheh 175
and Lecouvey extended this notion to subspaces in a field extension, here
we will introduce a notion from [2].
Let K ⊂ L be a field extension and A, B be n-dimensional K-subspaces
of L. Let A = {a1, . . . , an}, B = {b1, . . . , bn} be basis of A and B respec-
tively. It is said that A is matched to B if
aib ∈ A ⇒ b ∈ 〈b1, . . . , b̂i, . . . , bn〉
for all b ∈ B and i = 1, . . . , n, where 〈b1, . . . , b̂i, . . . , bn〉 is the hyperplane
of B spanned by the set B \ {bi}. Also it is said that A is matched to B if
every basis of A can be matched to a basis of B.
It is said that L has the linear matching property from K if, for every
n ≥ 1 and every n-dimensional K-subspaces A and B of L with 1 6∈ B, the
subspace A is matched to B. By this we mean linear matching property
for K-subspaces.
As we mentioned, the above notion was introduce by Eliahou and
Lecouvey in [2], where they proved that if K ⊂ L is a field extension
and [L : K] is prime, then L has linear matching property (see Theorem
5.3 in [2]). We extend this property to the family of field extensions and
introduce the notions of minimal and maximal linear matching properties.
2. Definitions and the main results
Definition 2.1. Let K be a field. We say K has minimal linear matching
property if there exists a finite field extension L of K, such that L has
linear matching property from K.
Definition 2.2. Let K be a field. We say K has maximal linear matching
property if for any positive integer n, there exists a field extension Ln of
K, such that [Ln : K]= n and Ln has linear matching property from K.
We shall prove the following results in section 5.
Theorem 2.3. Let K be a field which is not algebraically closed, then K
has the minimal linear matching property.
Theorem 2.4. Algebraic number fields have the maximal linear matching
property.
Theorem 2.5. Suppose that K is a field and has the maximal linear
matching property, then K is infinite.
To prove our main results, we will use Theorem 3.1 which can be
regarded as an improvement of the foundamental theorem of algebra.
176 On maximal and minimal linear matching property
In [6], Shipman gives an algebraic proof of the foundamental theorem of
algebra in special cases, but here we present a different proof which is
independent Shipman’s proof.
3. An improvment of the fundamental theorem of algebra
Theorem 3.1. Let K be a field such that every polynomial of prime degree
in K[x] has a root in K, then K is algebraically closed.
Proof. First, we claim there exists a prime p such that for any non-linear
irreducible polynomial f(x) ∈ K[x], p divides the degree of f(x). Suppose
that this claim is false, and p1, . . . , pn are prime divisors of the degree of
f(x), then there exists gi ∈ K[x] such that pi 6 | deg gi(x) and gi(x) is an
irreducible polynomials in K[x], where 1 ≤ i ≤ n.
Now set F (x) := fk0(x)gk1
1
(x) · · · gkn
n (x) where k0, k1, . . . , kn are non-
negative integers. It is clear that gcd(deg f(x), deg g1(x), . . . , deg gn(x)) =
1 and deg F = k0 deg f +k1 deg g1+· · ·+kn deg gn. By Dirichlet’s Theorem
on primes, since the ki’s are non-negative integers, we can choose k0, . . . , kn
such that deg F becomes a prime number. So F (x) has a root in K and
this is a contradiction. Therefore there exists a prime p such that p
divide the degree of every irreducible polynomials in K[x]. Now if L is
a field extension of K of degree p and α ∈ L \ K, then L = K(α) and if
f(x) ∈ K[x] is the minimal polynomial of α, then deg f(x) = p and f(x)
has a root in K and this is a contradiction, hence K has no field extension
of degree p. Let L be a Galois extension of K with [L : K] = pr · m where
r, m ∈ N, (m, p) = 1. By Galois fundamental theorem and Cauchy theorem,
there is an intermediate field L′, K ⊂ L′ ⊂ L such that [L : L′] = pr, then
[L′ : K] = m. If m > 1 we can choose α ∈ L′ \ K, and assume f(x) is the
minimal polynomial of α over K, then deg f(x)|m, also f(x) is irreducible,
then p| deg f(x), so p|m, a contradiction. Hence m = 1 and [L : K] = pr,
again by using Galois fundamental theorem and Cauchy theorem there
exists an intermediate field L′, K ⊆ L′ ⊂ L such that [L : L′] = pr−1,
then [L′ : K] = p, but since we proved that K has no field extension of
degree p, this is a contradiction. Thus K has no Galois extension and it
is algebraically closed.
Corollary 3.2 Let K be a field such that every polynomial of prime degree
in K[x] is reducible on K. Then K is algebraically closed.
M. Aliabadi, M. R. Darafsheh 177
4. Preliminary results about field extensions and linear
matching property
We use the following result from [4].
Theorem 4.1. Let L be a finite field of characteristic p > 0 where Zp is
embedded in L and [L : Zp] = n. Then for any divisor m of n, L has a
subfield with pm elements.
We also use the following result from [5] which is about field extensions
with no proper intermediate subfield.
Theorem 4.2. If K is an algebraic number field, then for every positive
integer n there exist infinitely many field extensions of K with degree n
having no proper subfields over K.
The following theorem was proved in [2], see also [1].
Theorem 4.3. Let K ⊂ L be a field extension. Then L has linear
matching property if and only if K ⊂ L has no proper intermediate
subfield with finite degree over K.
Now we are ready to prove the main results.
5. Proof of main results
Proof of Theorem 2.3
Proof. By Corollary 3.2 there exists an irreducible polynomial f(x) of
prime degree in K[x]. Now if L is the splitting field of f(x) over K, then
[L : K] is prime and by Theorem 4.3 L has the linear matching property
from K, so K has the minimal linear matching property.
Proof of Theorem 2.4
Proof. Let K be an algebraic number field. Then by theorem 4.2 for any
positive integer n, there exists an extension Ln of K with [Ln : K] = n
and this field extension has no proper intermediate subfield, then by
Theorem 4.3, Ln has the linear matching property from K, so K has the
maximal linear matching property.
Proof of Theorem 2.5
Proof. Let K be a finite field with |K| = pn and p a prime and n a
positive integer. Now let q and m be positive integers with n < q < m
178 On maximal and minimal linear matching property
and q|m. If L is an extension of K of degree m, then [L : Zp] = mn and by
Theorem 4.1, Zp ⊆ L has an intermediate subfield K ′ of degree pq. Now
since finite fields with the same cardinality are isomorphic, K ′ is a finite
proper intermediate subfield in the extension K ⊂ L with finite degree
over K, then by Theorem 4.3, L does not have linear matching property
from K, hence K does not have maximal linear matching property.
References
[1] S. Akbari, M. Aliabadi, Erratum to: Matching Subspaces in a field extension,
submitted.
[2] S. Eliahou, C. Lecouvey, Mathching subspaces in a field extension, J. Algebra. 324
(2010), 3420-3430.
[3] C.K. Fan, J. Losonczy, Matchings and canonical forms in symmetric tensors, Adv.
Math. 117 (1996), 228-238.
[4] D. S. Malik, Jhon N. Mordeson, M. K. Sen, Fundamentals of Abstract Algebra, Mc
GrawHill (1999).
[5] H. Marksaitis, Some remarks on subfields of algebraic number fields, Lithuanian
Mathematical Journal, Vol. 35. No. 2. (1995).
[6] J. Shipman, Improving of fundamental theorem of algebra, Math. Intelligencer 29
(2007), no.4, 9-14. 00-01 (12D05)
[7] E.K. Wakeford, On canonical forms, Proc. London Math. Soc. 18 (1918-1919),
403-410.
Contact information
M. Aliabadi Department of Mathematical Sciences, Sharif
University of Technology, Tehran, Iran
E-Mail: mohsenmath88@gmail.com
M. R. Darafsheh School of Mathematics, Statistics and Com-
puter Science, Colledge of Science, University of
Tehran, Tehran, Iran
E-Mail: darafsheh@ut.ac.ir
Received by the editors: 03.05.2012
and in final form 15.09.2012.
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