Exercises MAT2200 spring 2014 Ark 5 Rings and fields and factorization of polynomials This Ark concerns the weeks No. (Mar ) andno. (Mar ). Status for this week: On Monday Mar : Finished section 23(Factorization of Polynomials over a Field ) Section 26 (Homomorphisms and Factor Rings) Section 27 (Prime and Maximal Ideals). We stopped in the middle of the proof of Theorem 27.9. On Friday Mar we finish Section 27 and start on Part VI (Extension fields). The plans for the coming two weeks are as follows: Monday, Mar : Section 29 (Introduction to Extension Fields) Section 30 (Vector spaces) start on Section 31 (Algebraic Extensions). Friday, Mar : Exercises Section 31 (Algebraic Extensions). Monday, Mar : Section 36 (Sylow theorems) Section 37 (Application of the Sylow Theorems). Friday, Mar : Exercises + buffer The following are not part of the curriculum: Section 24(Noncommutative Examples) Section 25 (Ordered Rings and Fields) the paragraph in Section 27 on page 249 called Prime Fields Section 28 (Gröbner bases for Ideals) Section 32 (Geometric Constructions) Section 25(Ordered Rings and Fields) the last paragraph of Section 31 called Proof of Existence of an Algebraic Closure Section 34 (Isomorphisms Theorems) Section 35(Series of Groups) The exercises on this sheet cover the part of the curriculum in the sections from 20 to 27. They are ment for the groups on Friday Mar and Mar and Wednesday Mar and with the following distribution: Friday, Mar 21: No.: 3, 4, 8, 9, 11, 12, 13, 19, 20 Friday, Mar 28: No.: 22, 23, 25, 27, 29, 30, 33, 36, 37, 38 Wednesday, Mar 19: No.: 1, 2, 5, 6, 7, 10, 14, 15, 16, 17, 18, Wednesday, Mar 26: No.: 21, 24, 26, 28, 31, 32, 34, 35 Nilpotents, zero divisors and integral domains Oppgave 1. What are the nilpotent elements of Z 24?Whatarethezerodivisors? Oppgave 2. Which of the following rings are integral domains? R = Z 9 R = Z 11 R = Z 13 R = Z 15? For each R which is not an integral domain, describe all the zero divisors. 1
Oppgave 3. Show that if a and b are elements in the commutative ring R and a n = b m =0,then(a + b) n+m 1 =0. Conclude that the sum of two nilpotent elements is nilpotent. Hint: Use the binomial theorem, problem 30 Units and Euler s -function Oppgave 4. ( Section 20, No.: 2, 3 on page 189 in the book). Find the units in the rings Z 11 and Z 17.Showthattheyarecyclicandexhibitageneratorforeachofthem. Oppgave 5. Find the units in the ring Z 10 and show that Z 10 is cyclic of order 4. Oppgave 6. Find the units in the ring Z 9 and show that Z 9 is cyclic of order 6 generated by the residue class of 2. Oppgave 7. Find the units in the ring Z 8 and show that Z 8 ' Z 2 Z 2. Oppgave 8. Show that for any natural number n Euler s (n) n = Y p prime and p n (1 1 p ). -function satisfies Oppgave 9. Show that the group Z[i] of units in the ring Z[i] of Gaussian integers equals Z[i] = µ 4 = {±1, ±i}. (RecallthatZ[i] ={ a + ib a, b 2 Z } and i is the imaginary unit.) Squares and square roots Oppgave 10. List all the squares in the following rings R and decide in each case if 1 has a square root in R: R = Z 3, R = Z 5, R = Z 7. In which of the rings does 2 +2=0have a solution? Oppgave 11. Assume that p is an odd prime number. a) Show that ±1 are the only to elements in Z p with square equal to 1. Hint: Use that x 2 1=(x 1)(x +1)in any field. b) Regard the group homomorphism : Z p! Z p sending a to a 2.ShowthatKer = {±1}. Howmanyelementsarethereintheimageof? 2
Oppgave 12. (Wilsons theorem). Let G be an abelian group written multiplicatively. a) Assume that there is just one element a in G of order 2. Showthattheproductof all elements in G equals a; thatis Q x2g x = a. b) Apply this to G = Z p, where p is a prime, to show Wilsons theorem Hint: Problem 11 a) mightbeuseful. (p 1)! 1 mod p. Oppgave 13. In this exercise p denotes an odd prime. The aim of this exercise is to give a criterion for when an element a 2 Z p has a square root in Z p, i.e., for when there is a b 2 Z p with b 2 = a. a) Let : Z p! Z p be the group homomorphism given by (a) =a p 1 2. Show that Im = {±1}. Hint: Use Fermat s little theorem and exercise 11 a). b) Show that if a has a square root in Z p,thena 2 Ker. Hint: Fermat s little theorem. c) Show that a has a square root in Z p if and only if a p 1 2 =1. Hint: use problem 11 b). d) Show that 1 has a square root in Z p if and only if p 1 mod 4, i.e., if and only if p is of the form p =4k +1. Congruences and equations Oppgave 14. Solve the congruences 5x 7 mod 13 2x 6 mod 4 22x 5 mod 15 Oppgave 15. ( Section 19, No.: 1 on page 182 in the book). Find all solutions to x 3 2x 2 3x =0in Z 12. Oppgave 16. ( Section 19, No.: 2 on page 182 in the book). Solve the equation 3x =2in the fields Z 7 and Z 23. 3
Oppgave 17. (Basically Section 19, No.: 3 on page 182 in the book). Find all solutions of x 2 +2x +2=0in Z 5 in Z 6,andinZ 7 Oppgave 18. ( Section 19, No.: 4 on page 182 in the book). Find all solutions of x 2 +2x +4=0in Z 6. Oppgave 19. (Equations of the second degree). Let R be an integral domain in which 2 is invertible. Consider the quadratic equation x 2 + bx + c =0, (c) where a and b are elements in R. Showbythe(usual)procedureofcompletingthe square that (c) isequivalenttotheequation: (x + b/2) 2 = b 2 /4 c. Show that the equation (c) hasasolutioninr if and only if b 2 4c has a square root in R, and in that case, the solutions are given by the following (usual) formula: x = b ± p b 2 4c. 2 Oppgave 20. Show that the equation 2 I =0, where I denotes the identity matrix, has infinitely many solutions in the ring M [ (2)]R of real two by two matrices. Hint: Check that 0 1 = 1 0 is a solution and consider the conjugates AA 1 of. Factoring polynomials over finite fields Oppgave 21. ( Section 23, No.: 2 and 4 on page 218 in the book). In each of the cases below find the quotient and the remainder when dividing f() by g(): f() = 6 +3 5 +4 2 3 +2and g() =3 2 +2 3 in Z 7 [] f() = 4 +5 3 3 2 and g() =5 2 +2in Z 11 []. Oppgave 22. (Basically Section 23, No.: 9 on page 218 in the book). Show that 4 +4splits as a product of linear factors in Z 5 []. 4
Oppgave 23. Let f() = 3 +2 2 +2 +1. a) ( Section 23, No.: 10 on page 218 in the book). The polynomial f splits as a product of linear factors in Z 7 []. Findthefactorisation. Hint: 3 is a square in Z 7. b) Let p be an odd prime. Show that g() = 2 + +1splits a product of linear factors in Z p [] is and only if 3 is a square in Z p. c) Does f split as a product of linear factors over Z 11? Oppgave 24. ( Section 23, No.: 12 and 13 on page 218 in the book). In each of the following two cases, decide whether the polynomial f() is irreducible in Z 5 []. In case it is not, exhibit a factorization. Miscellaneous f() = 3 +2 +1 f() =2 3 + 2 +2 +2 Oppgave 25. Let N be a natural number. For any polynomial f 2 Z[], letf be the polynomial in Z N [] obtained by reducing the coefficients of f modulo N, i.e., if f() = P r i=0 a i i,weletf() = P r i=0 [a i] N i where [a] N denotes the residue class of a mod N. Show that f + g = f + g and fg = fg. Inotherwords,themapZ[]! Z N [] sending f to f is a ring homomorphism. Factoring polynomials over Z and Q Oppgave 26. Let A 2 Z be an integer with A 6= ±2. Showthat 4 + A 3 +1is irreducible in Z[]. Oppgave 27. Let f() = 4 + A 2 +1with A 2 Z. a) Assume that f() is not irreducible. Show that there is a factorization where d = ±1 and a 2 Z. f() =( 2 a + d)( 2 + a + d) b) Show that if f is not irreducible, then A =2 a 2 or A = 2 a 2 for some a 2 Z. Conclude that if A > 1, thenf is irreducible. c) Show that f() = 4 22 2 +1is irreducible in Z[]. d) Show that f() = 4 23 2 +1is not irreducible and find a factorisation of f in Z[]. 5
Oppgave 28. Assume that f() is a monic polynomial in Z[], i.e., the leading coefficient og f equals one. Let p be a prime, and let f() 2 Z p [] be the polynomial obtained by reducing the coefficients of f modulo p. a) Let f() = 4 +22 3 +12and p =11.Showthat f = 4 +1. b) Assume that f is irreducible in Z p []. Showthatf is irreducible in Z[]. Oppgave 29. Let p be an odd prime and let a and b be two nonzero elements in Z p. Show that either a, b or ab has a square root in Z p. Hint: Use Problem 13 c on Ark 7 where it is shown that an element a 2 Z p is a square if and only if a p 1 2 =1. Oppgave 30. a) Show that the polynomial f() = 4 10 2 +1is irreducible in Z[]. b) Show that we have the equality Hint: Square both sides. p 3 ± p 2= q 5 ± 2 p 6. c) Show that the following three equalities hold: 4 10 2 +1=( 2 2 p 2 1)( 2 +2 p 2 1) =( 2 2 p 3 +1)( 2 +2 p 3 +1) =( 2 (5 + 2 p 6))( 2 (5 2 p 6)) d) Why does this not contradict the unique factorization of polynomials with coefficients over the field Q( p 2, p 3)? The field Q( p 2, p 3) is defined as Q( p 2, p 3) = { a + b p 2+c p 3+d p 6 a, b, c, d 2 Q } e) Show that f is reducible in Z p [] for any prime p. Hint: Use the factorizations above and problem 29 with a =2and b =3. Ideals and factor rings Oppgave 31. ( Section 26, No.: 2 on page 243 in the book). Describe all ideals in Z 12.IneachcasedescribethefactorringZ 12 /I. Oppgave 32. If K is a field and : K! R is a homomorphism into a ring R with unity, show that f is injective. (We assume that (1) = 1). 6
Oppgave 33. Let : R! R 0 be homomorphism between the two rings R and R 0. a) Assume that is surjective. Prove that (I) is an ideal in R 0. b) Show by an example that the hypothesis of being surjective in a) is essential. c) Let J R 0 be an ideal in R 0.Showthattheinverseimage 1 (J) ={ x 2 R (x) 2 J } is an ideal in R. Oppgave 34. Assume that p is a prime and let b and c be elements in Z p.showthat Z p []/( 2 + b + c) is a field if and only if b 2 4c is not a square in Z p Oppgave 35. ( Section 27, No.: 6 and 7 on page 252 in the book). For which elements c in Z 3 are the following factor rings fields? Z 3 []/( 3 + 2 + c) Z 3 []/( 3 + c +1) Oppgave 36. Which of the following two rings are fields? Q[]/( 2 5 +5) Q[]/( 2 5 +6) Oppgave 37. Assume that R is a commutative ring with 1 and that I R is an ideal such that R/I is finite. Show that I is prime if and only if I is maximal. Oppgave 38. Show that A = Z 7 []/( 2 +1) is a field. Let i denote the residue class of. Showthati 2 = 1 and that any element in A can be written in a unique way as x = a + bi with a and b from Z 7. This motivates that we rebaptize A as Z 7 (i). How many elements does Z 7 (i) have? Is i asquareinz 7 (i)? Versjon: Tuesday, March 25, 2014 12:27:08 PM 7