Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation

On some classes of irreducible polynomials over finite fields

In this thesis we describe the work in literature on various aspects of the theory of polynomials over nite elds. We focus on properties like irreducibility and divisibility. We also consider existence and enumeration problems for irreducible polynomials of special types. After the introductory Chapter 1, we collect the well-known results on irreducibility of binomials and trinomials in Chapter 2, where we also present the number of irreducible factors of a xed degree k of xt due to L. Redei. Chapter 3 is on self-reciprocal polynomials. An in nite family of irreducible, self-reciprocal polynomials over F2 is presented in Section 3.2. This family was obtained by J. L. Yucas and G. L. Mullen. Divisibility of self-reciprocal polynomials over F2 and F3 is studied in Sections 3.3 and 3.4 following the work of R. Kim and W. Koepf. The last chapter aims to give a survey of recent results concerning existence and enumeration of irreducible polynomials with prescribed coefficients

Primitive polynomials with prescribed second coefficient

The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite fieldwith any coefficient arbitrarily prescribed. This has recently been provedwhenever n ≥ 9. It is also known to be truewhen n ≤ 3.We showthat there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e., that of xn−2) arbitrarily prescribed. In particular, this establishes the HMPC when n = 4. The lone exception is the absence of a primitive polynomial of the form x4 + a1x3 + x2 + a3x + 1 over the binary field. For n ≥ 6 we prove a stronger result, namely that the primitive polynomialmay also have its constant termprescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs

Prescribing coefficients of invariant irreducible polynomials

Let Fq be the finite field of q elements. We define an action of PGL(2,q) on Fq[X] and study the distribution of the irreducible polynomials that remain invariant under this action for lower-triangular matrices. As a result, we describe the possible values of the coefficients of such polynomials and prove that, with a small finite number of possible exceptions, there exist polynomials of given degree with prescribed high-degree coefficients

Some properties of generalized self-reciprocal polynomials over finite fields

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal polynomials and characterize the parity of the number of irreducible factors for a-self reciprocal polynomials over finite fields of odd characteristic

Existence problems of primitive polynomials over finite fields

This thesis concerns existence of primitive polynomials over finite fields with one coefficient arbitrarily prescribed. It completes the proof of a fundamental conjecture of Hansen and Mullen (1992), which asserts that, with some explicable general exceptions, there always exists a primitive polynomial of any degree n over any finite field with an arbitrary coefficient prescribed. This has been proved whenever n is greater than or equal to 9 or n is less than or equal to 3, but was unestablished for n = 4, 5, 6 and 8. In this work, we efficiently prove the remaining cases of the conjecture in a selfcontained way and with little computation; this is achieved by separately considering the polynomials with second, third or fourth coefficient prescribed, and in each case developing methods involving the use of character sums and sieving techniques. When the characteristic of the field is 2 or 3, we also use p-adic analysis. In addition to proving the previously unestablished cases of the conjecture, we also offer shorter and self-contained proof of the conjecture when the first coefficient of the polynomial is prescibed, and of some other cases where the proof involved a large amount of computation. For degrees n = 6, 7 and 8 and selected values of m, we also prove the existence of primitive polynomials with two coefficients prescribed (the constant term and any other coefficient)