Visual Algebra, Lecture 8.7: Finite fields скачать в хорошем качестве

Visual Algebra, Lecture 8.7: Finite fields

In the previous lecture, we learned that the quotient of a ring R by an ideal I is a field if and only if I is maximal. In this lecture, we'll learn about the structure of such field that are finite. We've already proven that finite integral domains are fields, but what do these finite fields look like, besides the familiar 𝔽ₚ={0,1,...,p-1}? We will see how to construct a finite field of order pⁿ as a quotient of the polynomial ring 𝔽ₚ[x] by an ideal (f(x)) generated by an irreducible polynomial of degree n. As an example, we will explicitly construct the additive and multiplicative Cayley tables of the finite fields of order 4 and 8, and we’ll explore the subring lattice of these fields, and of 𝔽₁₆. We’ll conclude by proving a few basic results, such as that every finite field has prime power order, and that the multiplicative group is always cyclic. Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule... ------------------------------------------------------------------------------------------------------------------------------------------------------ 0:00 Introduction 1:15 The characteristic of a field 3:39 Construction of 𝔽₄ 13:24 The Cayley tables and subring lattice of 𝔽₄ 16:45 Using software for finite field computations 21:13 The Cayley tables of 𝔽₈ 24:07 The subring lattices of 𝔽₄ and 𝔽₈ 28:21 The subring lattice of 𝔽₁₆ 30:29 The order of a finite field and its subfields 38:26 Finite multiplicative groups of a field