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The following glossary includes terminology covered thus far in class this semester. I have indicated page numbers in the textbook where these notions are introduced; also page numbers in the handouts distributed in class (and available on the course website). The relevant handouts are A. B. C. D. E. F. G. H. Three Examples of Proofs Mathematical Induction Review: Basic Notation and Properties of Integers Factorization in Rings More Factorization in Rings Basic Terminology and Results for Rings Polynomials, Power Series and Such Homomorphisms, Ideals and Quotients GLOSSARY Term / Concept mathematical proof mathematical induction well ordering Fundamental Theorem of Arithmetic Euclid’s Lemma Euclid’s Algorithm handout / pages A B B9, G9 B10, G8 G7 5 4 24, 58 23 18, 20 , , 0, , , C1 3-4, 75, 105 divides, divisibility, divisor, multiple greatest common divisor prime, composite coprime, relatively prime Division Algorithm quotient, remainder congruence C1, G5-6 C2, G7 C2 C2 C2, G5 C2 C3 17, 47 49 22, 27 27 15, 45, 197 15, 197 31 n, modular arithmetic C3 34-37 F[x], R[x], polynomial ring Mn(R), ring of n-by-n matrices over R C1, G4 C1 C2, E1 D2, E1 D2, E1 D3 D3, E1-2 42 79 144 [√𝑑 ] conjugate norm unique factorization units of a ring R, R× = U(R) Textbook pages 146 24 103, 317 associate reducible, irreducible, factoring migration of units operation (unary, binary, n-ary) ring zero element (additive identity) additive inverse commutative ring ring with identity (unity) isomorphism, isomorphic field zero divisor integral domain (domain) Euclidean domain subring quotient field D4 D4, E3,G8 D4 F1 F1 F1 F1 F3 F3 F4 F4 F4 F5, F6 F5 F5 F6, G10 , real quaternions F6-7 , octonions, nonassociative F7 skewfield (division ring) R[[t]], ring of power series coefficient degree of a polynomial zero (root) of a polynomial monic polynomial F(t), field of rational functions (rational expressions) F((t)), Laurent series proper subring homomorphism image, kernel automorphism ideal, left/right ideal principal ideal, generator principal ideal ring coset quotient ring well-defined map canonical homomorphism (natural homomorphism) epimorphism, monomorphism transcendental, algebraic First Isomorphism Theorem for Rings F7 G1 G4 G4 G5 G7 G10 G10 G13 H1 H1 H3 H9 H10 H11 H12 H12-13 H13 H14 H3 H12 H15 58, 141 55, 56 73 74 74 74 78 93 247 105 101 102 197 89 42 43 48 195 90 211 226 158 160 237 240 35, 237 241 493, 527 250