## Outline #6

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## Outline #6

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- Pages:
- 2
- School:
- Harvard University
- Course:
- Math 152 - Discrete Mathematics

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MATHEMATICS 152 FALL 2003 METHODS OF DISCRETE MATHEMATICS Outline 6 Rings and Fields especially Finite Fields None of this material is needed for the half hour quiz on Oct 9 which will cover outlines 1 through 5 and the first three homework assignments We define rings and fields provide basic examples construct more complex examples and study the structure of these objects 1 Define a ring as a set with two binary operations traditionally denoted and satisfying a set of axioms that Biggs summarizes as R1 R2 and R3 Expand R1 into A1 A5 R2 into M1 M2 and call R3 D A ring which satisfies M3 is said to be a ring with identity Axioms M5 and M4 are the extra algebraic properties that Biggs mentions at the top of p 197 A ring which also satisfies M4 is said to be a division ring and a ring which also satisfies M5 is said to be a commutative ring A ring that satisfies all eleven axioms is said to be a field Write out all the axioms explicitly Then by considering 1 1 a b and using the distributive law prove that for the case of a field the axiom that addition is commutative you probably called it A5 follows from the other axioms Sections 22 1 22 3 2 Show that the following examples satisfy the axioms above a the integers Z which form a commutative ring with identity b the congruence classes of integers modulo n denoted Zn which form a commutative ring with identity in all cases and a field when n is prime c the polynomials with real coefficients denoted R x which form a commutative ring with identity d the n n matrices with real entries denoted Mn R which form a ring with identity and a field in the special case n 1 Sections 22 1 22 3 and 22 4 3 Consider the more abstract rings of polynomials F x where F is any field In particular we are interested in the case where F Zp for a prime p State the Division Algorithm and the Euclidean Algorithm for polynomial rings Sections 22 4 22 6 1 4 Construct the equivalence classes of polynomials modulo q x for a specified polynomial q x of

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