Algebra 2 Exam 2019 PDF

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Algebra 2 Exam 2019...

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UNIVERSITY OF BRISTOL School of Mathematics ALGEBRA 2 MATH 21800 (Paper code MATH–21800)

May/June 2019 2 hours 30 minutes

This paper contains 4 (FOUR) questions. All FOUR answers will be used for assessment. Calculators are not permitted in this examination.

On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.

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Cont...

21800-19

Throughout, a “ring” means a commutative ring with 1. 1. Let R and S be rings. (a) (5 marks) Define what it means for a map ϕ : R → S to be a homomorphism. Define ker ϕ, the kernel of ϕ. (b) (5 marks) Define an ideal of R. Show that the kernel of a homomorphism is an ideal. (c) (4 marks) Let ϕ : R → S be a homomorphism with kernel I. Define ψ : R/I → S by ψ(x + I) = ϕ(x). Show that ψ is well-defined. (d) (2 marks) State the Fundamental Homomorphism Theorem. (e) (9 marks) Let I1 , I2 be two ideals of R such that I1 + I2 = R. Let S1 = R/I1 and S2 = R/I2 . Prove that there is a surjective homomorphism R → S1 × S2 . You may use without proof that the natural maps R → R/I1 , R → R/I2 are homomorphisms. 2. Give an example of each of the following, with an explanation: (a) (3 marks) An integral domain that is not a field. (b) (3 marks) A prime ideal that is not maximal. (c) (4 marks) Rings R ⊂ S such that R is an integral domain and S is not.

(d) (5 marks) A ring R with distinct elements r1 , r2 , r3 such that r12 = r22 = r32 = 1.

(e) (3 marks) A ring that is not a Principal Ideal Domain (PID). (f) (4 marks) An algebraic extension of fields of infinite degree. (g) (3 marks) A non-constructible real number. 3. (a) (2 marks) State (any version of) Gauss’ Lemma. (b) (6 marks) Which of the three polynomials x7 − 10x + 10, x2 − 6x + 8, x3 − x + 999999 in Q[x] are irreducible over Q? Justify your answers. (c) (3 marks) Let L ⊂ F be fields. Define the degree [F : L] and what it means for F/L to be an algebraic extension. √ √ √ √ 4 5 (d) (7 marks) Prove that the subfield K = Q( 2, 3 2, 2) of C does not contain 2. (e) (7 marks) Prove that every finite integral domain is a field. 4. (a) (5 marks) If k is a field, prove that the polynomial ring k[x] is a PID. (b) (8 marks) Let k = Z/pZ where p is a prime number. Show that there are exactly p + p(p − 1)/2 reducible monic quadratic polynomials in k[x], and deduce that there exists a field with p2 elements. (c) (7 marks) Show that x2 + x + 1 ∈ F2 [x] is irreducible. Let K = F2 [x]/(x2 + x + 1). Explain why K is a field. Write down its elements and the multiplication table of K . (d) (5 marks) Show that x2 + x + 1 ∈ F3 [x] is reducible, and exhibit an explicit zero divisor in the quotient ring F3 [x]/(x2 + x + 1).

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