Historically, finite commutative semifields have been studied as vector spaces or by the projective plane they coordinatise. Here we study finite commutative semifields from a purely algebraic standpoint. This dissertation is a thorough examination of finite commutative semifields via planar Dembowski-Ostrom (DO) polynomials. There is a one-to-one correspondence between finite commutative presemi-fields and planar DO polynomials. This result of Coulter and Henderson [14] is crucial to the development of this dissertation. We detail the development of this theory, the implications of this approach, search for new examples, and examine the combinatorial structures related to finite commutative semifields.;Chapter 1 outlines the necessary preliminary information, including a list of the known classical finite commutative semifields as well as a partial list of finite commutative semifields that have been discovered since work began on this dissertation. In Chapter 2 we detail the theory of this approach. The multiplication of a finite commutative semifield, R , can be represented by a bivariate polynomial, K( X, Y) = X ☆ Y. Making use of a convenient isotope, one so that x ☆ a = ax for all x ∈ R and a ∈ Nm , we develop the shape of the planar DO polynomial f(X) = K(X, X). Then for any a, b ∈ R we have a ☆ b = f( a + b) -- f(a) -- f(b). This form is then applied to determine what type of equivalences can exist among finite commutative semifields. Finally, we examine the isotopes that can exist between two planar DO polynomials of this shape and present restrictions on the shape of the istopisms.;The implications of Chapter 2 to the known semifields are explored in Chapter 3. Here we determine the general forms for the planar DO polynomials for the classical examples of finite commutative semifields. Additionally, we focus on how this theory can be applied to reprove the classification of commutative semifields whose dimension is four over their nucleus and two over their middle nucleus. This leads to some interesting results regarding planar DO polynomials. In Chapter 6 we exploit the results of Chapter 2 to program and run computer searches in the computer algebra program Magma for finite commutative semifields of order 35, 55, and 38. We outline the methods used and the results. In each case, we discover a new example of a finite commutative semifield.;The next two chapters explore the combinatorial structure closely linked to finite commutative semifields. In Chapter 5 we examine the connection with projective planes and difference sets. The new examples described in Chapter 3 yield new projective planes, and the new example of order 35 yields a new skew Hadamard difference set. Chapter 6 focuses on linear codes. We detail the construction of a linear code from the image of a planar DO polynomial and present some interesting principles regarding the resulting linear codes. Finally we end with Chapter 7 in which we discuss some of the problems left open.
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