Linear Algebra over Finite Fields

Aishah Ibraheam Basha

Finite fields play a crucial role in algebra. Indeed, finite fields are the basic representation used to solve many integer problems. On the other hand, many elementary courses in linear algebra focus on studying infinite fields and it is always assumed to be either the real or the complex. In my dissertation, we discuss linear algebra over finite fields, namely we focus on Fqm , where q is a prime power. The most interesting question is:What changes for linear algebra over a finite field?\nThis question asks which standard results from linear algebra change when we go from\nan infinite field to a finite field Fqm, where q is a prime power. How much linear algebra\ncan be done over a finite field Fqm . Much of linear algebra may be formulated and remains\ncorrect for finite field Fqm , but some of results change and they are not longer true over finite\nfields. Most of linear algebra essentially only depends on the fact that you are working over\na field. But when you are working with a finite field Fqm , you often don’t have a notion of\ndistance, angles, slopes, etc. All our results in this dissertation depends on one fact which is\nn\nthere are nonzero vectors over finite fields Fqm , as vector spaces over Fqm , such that the norm\nof these vectors is zero. This problem makes a lot of linear algebra over finite fields fail. In \nthis paper, we present all results from linear algebra and determine which results are still true with reproving and explaination. We present briefly some topics over finite fields that are needed for linear algebra such as conjugates and norms of elements over finite field Fqm . Then we summerize some basics of linear algebra with emphasis on vectors and matrices. Finally, we consider the most important application of linear algebra over finite fields, the numerical range of a matrix A over Fq2 . We define a new numerical range over Fq2 . Then we explain why do we need this new definition with proving important results using this new definition.

Linear Algebra over Finite Fields

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