Lectures on Abstract Algebra [Lecture notes] by Richard Elman

By Richard Elman

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1 without using the Fundamental Theorem of Arithmetic. 2. Let F = R, C or Q [or any field, or even any ring, cf. 3]. Let F [t] be the set of polynomials with coefficients in F with the usual addition and multiplication. ) State and prove the analog of the Division Algorithm for integers. (Use your knowledge of such division. ) What can you do if you take polynomials with coefficients in Z? 3. Prove the following modification of the Division Algorithm: If m and n are two integers with m nonzero, then there exist unique integers q and r satisfying 1 1 n = mq + r with − |m| < r ≤ |m|.

1). eG ϕ(eG ) = ϕ(eG ) = ϕ(eG eG ) = ϕ(eG )ϕ(eG ). By Cancellation, eG = ϕ(eG ). (2). eG = ϕ(eG ) = ϕ(aa−1 ) = ϕ(a)ϕ(a−1 ). Similarly, eG = ϕ(a−1 )ϕ(a). (3), (5) are left as exercises. (4). We have ϕ(a) = ϕ(b) if and only if eG = ϕ(a)ϕ(b)−1 = ϕ(a)ϕ(b−1 ) = ϕ(ab−1 ) if and only if ab−1 ∈ ker ϕ. So ker ϕ = {eG } if and only if ϕ(a) = ϕ(b) implies a = b. 8. 1. Let G and H be groups, then the map ϕ : G → H given by x → eH is a group homomorphism, called the trivial homomorphism. 2. Let H be a subgroup of G.

Let H be a subgroup of G. Define ≡ mod H by if a, b ∈ G then a ≡ b mod H if and only if b−1 a ∈ H.