CC BY
10
УДК 512.554
Minimal Polynomials in Finite Semifields
Olga V. Kravtsova*
Institute of Mathematics and Computer Sciences Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 13.2.2018, received in revised form 23.04.2018, accepted 20.06.2018 We consider the classical notion of a minimal polynomial and apply it to investigations in finite semi-fields. A proper finite semifield has non-associative multiplication, that leads to a number of anomalous properties of one-side-ordered minimal polynomials. The interrelation between the minimal polynomial of an element and the minimal polynomial of its matrix from the spread set is described and illustrated by some semifields of orders 16, 32 and 64.
Keywords: semifield, right-ordered degree, right-ordered minimal polynomial. DOI: 10.17516/1997-1397-2018-11-5-588-596
Introduction
A finite semifield is an algebraic structure generalizing the notion of a finite field. Unlike any field, the multiplicative law in a proper semifield is non-associative. The absence of associativity even in a finite semifield leads to it having a number of specific properties, which are poorly studied.
The investigation of finite semifelds originated as a classical part of algebra in the works of Dickson [1] and Albert [2] at the start of the 20th century. The complete review is presented by Johnson at al. in Handbook [3]. At present the sphere of semifield studies is concerned with the calculation of element orders and automorphism groups [4]. The present paper is devoted to the application of classical algebraic concepts, such as minimal polynomials, in the specific area of finite semifields. We describe the interrelation between one-side-ordered minimal polynomials of the elements and minimal polynomials of its matrix from the spread sets of a semifield and the opposite semifield. This interrelation is illustrated by the examples of some semifields of orders 16, 32, 64. The research methods are closely related to linear spaces and spread sets, they combine both a theoretical and a computer approaches.
1. Definitions
According to [5], a semifield is a set W with two binary algebraic operations + and * such that:
1) {W, +) is an abelian group with neutral element 0;
2) {W*, *) is a loop (W* = W \ {0});
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3) both distributivity laws hold, a * (b + c) = a * b + a * c, (b + c) * a = b * a + c * a for all a,b,c € W.
A semifield W contains subsets Nr, Nm, N which are called right, middle and left nuclei respectively:
Nr = {n € W I (a * b) * n = a * (b * n) Va,b € W}, Nm = {n € W I (a * n) * b = a * (n * b) Va, b € W}, N = {n € W I n * (a * b) = n * (a * b) Va,b € W}. The intersection N = N n Nm n Nr is called the nucleus of semifield W and its subset
Z = {z € N I z * a = a * z Va € W}
is the center of W. The center and all nuclei of a finite semifield are its subfields and the semifield is a linear space over any of them. Therefore, the order of finite semifield is the prime number degree pn.
Any finite semifield may be constructed on the basis of a linear space over an appropriate finite field. Let W be a n-dimensional linear space over the field Fp, d is a bijective mapping from W to GLn(p) U {0} such that:
1) e(u + v) = 6(u) + 6(v) Vu, v € W,
2) 9(0,0,..., 0) = 0 is zero matrix, 0(1, 0,..., 0) = E is identity matrix. Define the multiplication law on W by the rule
u * v = u0(v), u,v € W,
then (W, +, *) is a semifield of order pn. Denote it W = W(n,p,0). The multiplicative neutral element 0-1 (E) is denoted as e. The image R = {0(u) | u € W} is called a spread set (see [5]).
Note that the center Z of a semifield is usually used as a basic field to construct a semifield. Nevertheless, it is more convenient to consider a linear space W and a spread set R over the prime subfield Fp.
The product of m multipliers is said to be m-th degree of a fixed element v € W*, if every multiplier coincides with v. The smallest integer m ^ 1 such that there exists the m-th degree of v, which is equal to the identity, is called the order of v and denoted by |v|. The set of orders of all elements is called the spectrum of multiplicative loop W*.
Similarly, using the right-ordered and the left-ordered m-th degrees
vm) = vm-1) * v, v(m = v * v(m-1, v1 = v = v(1,
we define the right order IvIr and the left order |v|; of v and the right and the left spectra of W* respectively.
Remind now the classical definition of a minimal polynomial for an element in a finite field and the main properties of minimal polynomials, according to [6].
Let be a field of order qn and a € . The monic polynomial M(x) € Fg [x] of a minimal degree such that M(a) = 0 is called the minimal polynomial of an element a over Fg.
Theorem 1.1. Let a € , a is of d-th degree over Fg and M (x) is a minimal polynomial of the element a over Fg.
(i) M(x) is irreducible over Fg and deg M = d is a factor of n.
(ii) For any polynomial f € Fg [x] the equality f (a) = 0 holds iff MIf.
(iii) If a is primitive element, then deg M = n.
(iv) The roots of M (x) are exactly a, aq,..., aq , and M (x) is a minimal polynomial for these elements.
(v) If f (x) is monic irreducible polynomial from Fq [x] and f (a) = 0, then f = M.
(vi) M (x) is a factor of xq — x and of xq — x.
We attempt to apply classical notion of a minimal polynomial to study finite semifields. Consider the polynomial f (x) G Fp[x],
f (x) = cmxm + cm-ixm~1 +----+ C2x2 + cix + co, Ci G Fp, i = 0,1, ... ,m.
For any element a G W, define the right- and the left-ordered value of a polynomial f (x):
f (a)) = cmam) + cm-1am-1) +-----+ c2a2 + cia + coe,
f ((a) = cma(m + cm-ia(m-1 +-----+ c2a2 + cia + coe.
Here as) and a(s are the right- and the left-ordered degrees of an element a. The product of the coefficient c G Fp to element a G W equals to the sum of c items coinciding with a, for c = 0, and equals zero for c = 0.
Example. Let f (x) = x3 + x2 + x + 1 G F2[x], a G W(n, 2, 0), then
f (a)) = a3) + a? + a + e = a? * a + a? + a + e = a0(a)0(a) + a0(a) + a + e, f ((a) = a(3 + a? + a + e = a * a2 + a? + a + e = a0(a0(a)) + a0(a) + a + e.
Evidently, in the case of degree < 2 the right- and the left-ordered values f (a)) and f ((a) are equal. Note that (fg)(a)) is not equal to f (a)) * g(a)) in general. If f (a)) = 0 then for any polynomial g(x) holds (fg)(a)) = 0, but inverse is not true: an equality (fg)(a)) = 0 does not imply f (a)) = 0 or g(a)) = 0 (similarly for left-ordered values).
The right-ordered minimal polynomial of an element a G W(n,p,0) is said to be a monic polynomial nra(x) G Fp[x] of minimal degree such that tira(a)) = 0. The left-ordered minimal polynomial nla(x) is defined likewise. Here we consider polynomials only over the prime subfield Fp, but it is possible to apply the results to the case of the center Z of a semifield as a basic field.
2. Properties
Some properties of one-side-ordered minimal polynomials in a finite semifield correspond to similar results in finite fields.
Lemma 1. If a = 0 then mra(0)) = 0, mla((0) = 0, i.e. x is not a factor of mra(x) and mla(x).
Proof. If mra(x) = c0xm + ■ ■ ■ + cm-1x = (c0xm-i + ■ ■ ■ + cm-1) ■ x, then
mra(a)) = (co am-1) + ■■■ + cm-i) * a = 0, coam-1) + ■■■ + cm-i = 0,
that contradicts the definition of a right-ordered minimal polynomial. For the left-ordered minimal polynomial the result can be obtained similarly. □
Theorem 2.2. For any polynomial f (x) G Fp[x] holds:
(i) f (a))=0 iff mra(x)\f (x);
(ii) f ((a)=0 iff mla(x)\f (x).
Proof for the right-ordered polynomial. If g(x) = mra(x) ■ x then g(a)) = mra(a)) * a = 0 * a = 0, so for
f(x) = mra(x) ■ (doxk + ■■■ + dk) € Fp[x]
holds f (a)) = 0. And inverse, let f (a)) = 0, make division with residual:
f (x) = mra(x)q(x) + r(x),
where r(x) is zero polynomial or degr < degmra. Then
f (a)) = (mra ■ q)(a)) + r(a)) = 0 + r(a)) = 0,
and minimality of mra(x) implies r(x) = 0. □
Evidently, this result leads the next theorem.
Theorem 2.3. The right-(left-)ordered minimal polynomial of an element a is a factor of the polynomial xk — 1, where k is right (left) order of a, k = |a|r (k = IaIi).
The right- or the left-ordered polynomial of an element a € W is not necessarily irreducible. Next lemma is a corollary from Theorem 2.1.
Lemma 2. If K ^ Fpm is a subfield of a semifield W then for a € K right-ordered polynomial of a equals to the left one, mra(x) = mla(x), it is an irreducible polynomial of order sIm.
The inverse result is not true: if a one-side-ordered polynomial is irreducible, then the element does not necessarily lie in a subfield (see Section 4).
Lemma 3. If ^ is an automorphism of a semifield W then a and av have the same right- and left-ordered minimal polynomials,
mTaV (x) = ma (x) m'a-P (x) = mla (x) .
This result seems evident but note that the automorphism group of a finite field Fp™ is a cyclic group generated by the automorphism x ^ xp. In the case of a semifield of order pn this mapping is not an automorphism in general and mrap (x) may differ from mra(x).
Compare one-side-ordered polynomials of an element a € W = W(n,p,0) to minimal polynomial (in a classical sense) of a matrix from the spread set, A = 0(a).
Lemma 4. For any polynomial f (x) € Fp[x], f (0) = 0, any element a € W(n,p,0) and correspondent matrix A = 0(a) the equality f (A) = 0 implies f (a)) = 0.
Proof. Let f (A) = 0. f (0) = 0, so we can assume c0 = —1 without loss of generality and
cmAm + cm-1Am-1 + ■■■ + c2A2 + c1A = E,
A(cmAm-1 + cm-1Am-2 + ■■■ + c2 A + c1E) = E. Multiply the left and the right parts of equality by the identity of a semifield W:
eA(cmAm-1 + cm-1Am-2 + ■■■ + c2 A + c^E) = eE,
a(cmAm-1 + cm-1 Am-2 + ■■■ + c2A + cxE) = eE, cmam) + cm-1 am-1) +-----+ c2a2 + c1a = e
and f (a)) =0. □
Note that inverse is not true in general, i. e. equality f (a)) = 0 does not imply f (A) = 0. Indeed, rewrite the equality
a(cmAm-i + cm-iAm-2 + ■■■ + c2A + ciE) = eE
in the form aB = e, then a = eB-1. As a = eA = e0(a) then e(B-1 — A) = 0 and it implies only that the first row of A equals to the first row of B-1, but not the equality of matrices A = B-1.
The similar result for the left-ordered polynomial is not true in general (a counter-example can be seen in Section 4).
The condition f (0) =0 in the text of the lemma is not essential because the absence of zero divisors in a semifield.
Therefore we have the correspondence between minimal polynomials.
Theorem 2.4. If a G W(n,p, 0) and A = 0(a), then the right-ordered minimal polynomial of an element a is factor of the minimal polynomial of the matrix A.
The bijective mapping ^ from the semifield {W,, *) to the semifield {V, o) is an anti-isomorphism if
(x * y)* = y* o x* Vx, y G W.
Lemma 5. If ^ is an anti-isomorphism from W to V, then for any element a G W* the right-ordered minimal polynomial mra(x) equals to the left-ordered minimal polynomial of the element a* G V.
Corollary 1. If the semifield W is anti-isomorphic to itself then for any element a G W* the right-ordered-minimal polynomial coincides with the left-ordered one.
Lemma 6. If ^ is an anti-isomorphism from W = W(n,p,0) to V = V(n,p,T) then for any a G W* the left-ordered minimal polynomial mla(x) is a factor of the minimal polynomial of the matrix t(a*).
3. Example
Illustrate previous results by the example of a semifield of order 16. It is known that there exist 23 pairwise non-isomorphic semifields of order 16 [4]. Let us consider one of them that is represented by a 4-dimensional linear space.
Let W be a 4-dimensional linear space over F2,
W = {x = (xi, x2, x3, x4) \ xi G F2, i = 1,..., 4}.
We define the map 0 : W ^ GLA(2) U {0} by the rule
1\ 0 1 .
1
16 where
x * y = x ■ 0(y). The vector e = (1, 0,0,0) is an identity in this semifield.
Q(x\, x2, X3, X4) = xiE + x2
/0 1 0 0 0 0 1 0 0 0 0
0 0 0 1 + x3 0 0 1 1 + x4 1 0 1
0 0 1 1 1 0 1 0 1 1 1
\1 0 1 1 0 1 0 1 1 1 0
Then, R = [9(x) | x G W} is a spread set and {W,, +, *) is a semifield of order
Except for direct search of variants, we can use the following method to construct a one-side-ordered minimal polynomial of elements. Let e1y e2,..., en be the base of a linear space W = W (n,p,9) over Fp. Write down the right-ordered degrees e, a, a2, a3),..., an) as linear combinations of basic vectors:
a%) otij ej,
here a.ij G Fp
i = 0,1,... ,n, j = 1,... ,n. Then make a matrix (aj) and lead it to a trapezoid form. The zero row corresponds to a linear combination of right-ordered degrees of a which equals to zero, i. e. to the right-ordered minimal polynomial of a.
For example, let's consider e = (1,0,0, 0), b = (0, 0,1,1), b2 = (1,1, 0,1), b3) = (0,0,1, 0), b4) = (0,1,0,1) in the semifield W = W(4, 2, d), then
/1 0 0 0 e\
0 0 11 b
110 1 b2
0 0 10 b3)
^0 1 0 1 b4)J
1 0 0 0 e
0 10 1 b2 + e
0 0 1 1 b
0 0 0 1 b3) + b
^0 0 0 0 b4) + b2 + ej
So, the right-ordered minimal polynomial of the element b is mrb(x) = x4 + x2 + 1 = (x2 + x + 1)2. Note that it is not irreducible. All one-side-ordered polynomials and all one-sided orders for a G W \ {0, e} are presented in Tab. 1. The last column shows the minimal polynomial of the matrix A = 0(a).
Table 1. Information on W(4, 2, 9)
a 14 \a\r \a\ mla(x) ma(x) m A (x)
(0,0,1, 0) 3 3 3 x2 + x + 1 x2 + x + 1 (x2 + x + 1)2
(1, 0,1, 0) 3 3 3 x2 + x + 1 x2 + x + 1 (x2 + x + 1)2
(0,1,0,1) 3 3 3 x2 + x + 1 x2 + x + 1 x2 + x + 1
(1,1,0,1) 3 3 3 x2 + x + 1 x2 + x + 1 x2 + x + 1
(0,0,0,1) 5 6 4 x4 + x3 + x2 + x + 1 (x2 + x + 1)2 (x2 + x + 1)2
(1,0,1,1) 5 6 4 x4 + x3 + x2 + x + 1 (x2 + x + 1)2 (x2 + x + 1)2
(0,1,0, 0) 5 15 5 x4 + x3 + x2 + x + 1 x4 + x + 1 x4 + x + 1
(0,1,1, 0) 5 15 5 x4 + x3 + x2 + x + 1 x4 + x + 1 x4 + x + 1
(0,1,1,1) 6 15 6 (x2 + x + 1)2 x4 + x + 1 x4 + x + 1
(1, 1, 1, 1) 6 15 6 (x2 + x + 1)2 x4 + x + 1 x4 + x + 1
(0,0,1,1) 15 6 5 x4 + x3 + 1 (x2 + x + 1)2 (x2 + x + 1)2
(1,0,0,1) 15 6 5 x4 + x3 + 1 (x2 + x + 1)2 (x2 + x + 1)2
(1,1,0, 0) 15 15 5 x4 + x3 + 1 x4 + x + 1 x4 + x + 1
(1,1,1, 0) 15 15 5 x4 + x3 + 1 x4 + x + 1 x4 + x + 1
The elements of order 3, together with 0 and e, form two subfields of order 4. The right- and the left-ordered minimal polynomials of these elements are equal to the irreducible polynomial x2 + x +1. Note that we have the polynomial of degree 2 that has four distinct roots. Similarly, the polynomial x4 + x + 1 has 6 roots in W. In all cases, the right-ordered minimal polynomial of an element is a factor of the matrix minimal polynomial. Nevertheless, the left-ordered minimal polynomial is a factor of the matrix polynomial only for elements from the subfields.
The semifield W(4,2,0) admits unique nontrivial automorphism y of order 2,
0\
0 0 .
1
One can prove that for any element a G W the right- and left-ordered minimal polynomials of its image a* coincide with mra(x) and mla(x) respectively. For example, (0, 0, 0,1)* = (1,0,1,1) (see Tab. 1).
The mapping x ^ x2 is not injective on W (compare with field F16). Indeed, (0,0,0,1)2 = = (0, 1, 0, 1)2 = (1, 1, 0, 1) and one-sided minimal polynomials of (0, 0, 0, 1), (0, 1, 0, 1) and (1, 1, 0, 1) are not the same.
For any ring (or semifield) R = (R, +, ■) the opposite ring Rop = (R, +, o) is determined by a o b = b ■ a (a,b G R). It is clear that the rings Rop and R are anti-isomorphic. Now let V = V(4, 2,t) = Wop, then we obtain the matrices t(ei) from the equality ei0(ej) = ejt(ei), i,j = 1,..., 4. So
/0 0 0 1\ 10 11 0 10 1. 1101
Direct calculations of minimal polynomials for all elements a G V = Wop serve the illustrations for Lemmas 5, 6: the left-ordered minimal polynomial of a G Wop coincides with the right-ordered minimal polynomial of a G W and inverse. Further, if A = 0(a) and Aop = t(a), then the left-ordered minimal polynomial of a G W is a factor of mA°p (x). For example, the minimal polynomial of t(1,1,1,0) is x4 + x3 + 1 (see Tab. 1 and compare to the last row).
4. Primitivity
In 1991 Wene [7] wrote the hypothesis: any finite semifield W is right or left primitive, i. e. the loop W* is a set of right- or left-ordered degrees of some element in a semifield W. In 2004 Rua [8] gave a counter-example to Wene's conjecture, using a Knuth semifield of order 32. This commutative Knuth-RUa semifield is neither right nor left primitive. The second counterexample is Hentzel-Rua semifield of order 64, which was constructed in 2007 [9]. Now the primitivity investigations are completed for all semifields of orders up to 125. There exist only two semifields of order ^ 125 (as can be seen above), which are neither left nor right primitive. Note that the counter-examples of odd order are still unknown.
The investigations of primitivity are based on the properties of a spread set. It is known that for any finite semifield W with the center Z(W) ^ Fq and the spread set £ the characteristic polynomial for any matrix from £ \ {XE \ X G Fq} has no linear factors. The following theorem serves as the main tool, which was used in [9].
Theorem 4.5. If W is a finite semifield of dimension n over its center Z(W) = Fq, then w G W is a left primitive element of W iff the characteristic polynomial of a linear map Lw : W ^ W, given by Lw (x) = w * x, is an irreducible primitive polynomial of degree n over Z(W).
: (xi, x2, x3, x4) ^ (xi, x2, x3, x4)
100 0 1 1 0 0 1 101
t(xi, x2, x3, x4) = xiE + x2
0 1 0 0 0 0 1 0
0 0 0 1 + x3 0 0 1 1
0 0 1 1 1 0 1 0
1 0 1 0 1 1 1 1
+ x4
Note that this result is formulated for a vector-column. In the case of a vector-row we must change the mapping Lw to Rw : x ^ x * w. For the semifield W(4, 2, 0) above we have 6 right primitive elements (that is of right order 15) and 4 left primitive elements. Their matrix minimal polynomial is an irreducible polynomial of degree 4.
As it is stated in [4], Knuth-Rua semifield of order 32 and Hentzel-Rua semifield of order 64 have no elements of one-sided order 31 and 63 respectively. The following definition gives a weakening for the Wene's hypothesis.
Any finite semifield W, which is n-dimensional over its center Z(W), is said to be right-cyclic if for some element a G W the semifield W has Z(W)-base
{e, a, a2),...,an-1)}.
The next result seems evident, but we prove it for completeness. Theorem 4.6. Any right-primitive semifield is also right-cyclic.
Proof. Let W be n-dimensional over its center and W have a right primitive element a, then the minimal polynomial of the matrix 0(a) is an irreducible polynomial of degree n. As the right-ordered minimal polynomial of an element a is a factor of matrix one then mra(x) is also of degree n. Thus, the elements e, a, a2, a3),..., an-1) are linear independent and as such form the base of the n-dimensional linear space. □
Both non-primitive semifields of order 32 and 64 are left-cyclic and right-cyclic. These semi-fields contain the elements with one-side-ordered polynomials of degree 5 and 6 respectively (see Tab. 2 and Tab. 3).
Table 2. Information on Knuth-Ruua semifield of order 32
\a\i = \a\r = |a| mla(x) = mra(x) = mA(x)
5, 8 x5 + x4 + 1 = (x2 + x + 1)(x3 + x + 1)
6, 10 x5 + x + 1 = (x2 + x + 1)(x3 + x2 + 1)
7 x5 + x4 + 1 or x5 + x + 1
Table 3. Information on Hentzel-Rua semifield of order 64
\a\i = \a\r \a\ ma(x) = mra(x) mA(x)
7 6 x6 + x5 + x4 + x3 + x2 + x + 1 = = (x3 + x + 1)(x3 + x2 + 1) x6 + x5 + x4 + x3 + x2 + x + 1 = = (x3 + x + 1)(x3 + x2 + 1)
12 7 x6 + x5 + x3 + x + 1 = = (x2 + x + 1)3 x6 + x5 + x3 + x + 1 = = (x2 + x + 1)3
15 5 x4 + x +1 x6 + x5 + x4 + x3 + 1 = = (x4 + x + 1)(x2 + x +1)
6 6 x4 + x2 + 1 = = (x2 + x + 1)2 x6 + x5 + x3 + x + 1 = = (x2 + x + 1)3
7 7 x3 + x +1 or x3 + x2 + 1 x6 + x2 + 1 = (x3 + x + 1)'2 or x6 + x4 + 1 = (x3 + x2 + 1)2
3 3 x2 + x +1 x2 + x +1
Further information on the structure of exceptional semifields can be found in [4]. This work was funded by Russian Foundation for Basic Research, project 16-01-00707.
References
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[3] N.L.Johnson, V. Jha, M. Biliotti, Handbook of finite translation planes, Pure and applied mathematics. Chapman&Hall/CPC, 2007.
[4] V.M.Levchuk, O.V.Kravtsova, Problems on structure of finite quasifields and projective translation planes, Lobachevskii Journal of Mathematics, 38(2017), no. 4, 688--698.
[5] D.R.Hughes, F.C.Piper, Projective planes, Springer-Verlag New-York Inc., 1973.
[6] R.Lidl, G.Pilz, Applied Abstract Algebra, Springer-Verlag New York, 1984.
[7] G.P.Wene, On the multiplicative structure of finite division rings, Aequationes Math., 41(1991), 791-803.
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Минимальные многочлены в конечных полуполях
Ольга В. Кравцова
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Используется классическая техника минимальных многочленов для исследования конечных полуполей. Отсутствие ассоциативности умножения приводит к аномальным свойствам односторонне-упорядоченных минимальных многочленов. Описана связь минимального многочлена элемента полуполя и его матрицы из регулярного множества. Результаты иллюстрированы примерами некоторых полуполей порядков 16, 32 и 64.
Ключевые слова: полуполе, правоупорядоченная степень, правоупорядоченный минимальный многочлен.
