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1 Modern Computer Algebra Exercises to Chapter 25: Fundamental concepts 11 May 1999 JOACHIM VON ZUR GATHEN and JÜRGEN GERHARD Universität Paderborn

2 25.1 Show that any subgroup of a group G contains the neutral element 1 of G Show that cyclic groups are commutative Let G = GL 2 (R ) be the group of invertible 22 matrices over R. Show that 1 1 U = A 2 G:A = 0 0 is a subgroup of G Let E 12 = fe i j=6 :0 j < 12g C, where i = p 1 is the imaginary unit. (i) Show that a 12 = 1 holds for all a 2 E 12. (That is why the elements of E 12 are called the 12th roots of unity.) (ii) Mark the elements of E 12 on the unit circle in the complex plane. (iii) Show that E 12 is a commutative group with respect to the multiplication of complex numbers. (iv) Show that the set E 4 = fe ik=2 :0 k < 4g of 4th roots of unity is a subgroup of E 12. Highlight the elements of E 4 in your drawing. (v) Determine all (left) cosets with respect to E 4. (vi) Set up the multiplication table of the factor group E 12 =E Show that the set E = fz 2 C :9n 2 N 1 z n = 1g C of all complex roots of unity is a subgroup of (C ;) S 4 is the set of all bijective maps (permutations) from f1; : : : ;4g to itself. We represent an element 2 S 4 as ((1)(2)(3)(4)), and examine the following subset: V = f(1234); (4321); (2143); (3412)g: (i) Draw cycle diagrams for the elements of V. (ii) Show that V together with the composition of maps is a subgroup of S 4, and compute the multiplication table of V Let G = fx 2 Q :0 x < 1g. (i) Show that G together with the operation x + y falls 0 x + y < 1 x y = x + y 1 falls x + y > 1 is a commutative group with infinitely many elements. (ii) Show that all elements of G have finite order. (Hint: express x 2 G as a fraction.) (iii) Prove that G is isomorphic to the factor group Q =Z with respect to addition. c 1999 JOACHIM VON ZUR GATHEN and JÜRGEN GERHARD 2

3 Exercises to Chapter (i) Show that the set S R = f f : R! R ; f bijectiveg is a group with respect to the composition of maps. Is this group commutative? (ii) For a;b 2 R let f a;b : R! R be the function given by f a;b (x) = ax +b. Show that the set G = f f a;b :a;b 2 R ; a 6= 0g is a subgroup of S R. Is G commutative? (iii) Show that H = f f 1;b :b 2 R g is a commutative subgroup of G (i) Are S 3 and (Z 6 ; +) isomorphic? Explain your answer. (ii) Show that ': Z! Z with '(x) = 3x is a homomorphism with respect to addition in Z. (iii) Let R >0 = fx 2 R :x > 0g. Show that ': R >0! R >0 with '(x) = 3x is not a homomorphism with respect to multiplication in R Which of the following maps are group homomorphisms? Determine the kernel and the image of all homomorphisms. (i) ': (R ; +)! (C ;), '(x) = e ix. (ii) ': (C ;)! (R ;), '(x) = jxj. (iii) ': (Z 17 ;)! (Z 17 ;), '(x) = x2. (iv) ': (Z; +)! (Z; +), '(x) = x (v) h: (Z 3 ; +)! (Z 4 ; +), h(0) = 0, h(1) = 1, h(2) = Let U be a subgroup of the group G. Show that U U = U, where U U = fu 1 u 2 :u 1 ;u 2 2 Ug Let p 2 N be prime and U < Z p a subgroup of the additive group (Z p ; +) with U 6= f0g. Show that U = Z p Prove: (i) If the order of a group G is prime, then there exists a primitive element g in G, such that hgi = G. (ii) If H and K are subgroups of the finite group G and gcd(#h;#k) = 1, then H \ K = feg. (Hint: Show first that H \ K is a subgroup of G.) Show that any cyclic group G of order n is isomorphic to Z n (with addition modulo n). Thus there is essentially only one cyclic group of order n Let ':G! H be a homomorphism of multiplicative groups. Show that ker' is a normal subgroup of G (this means that it is a subgroup and g 1 ag 2 ker' for all g 2 G and a 2 ker') and '(G) is a subgroup of H. What is the analogous statement for rings? Let G be a group. Prove: (i) G is commutative if and only if the inversion mapping x 7! x 1 is a group homomorphism. (ii) G is commutative if and only if the squaring mapping x 7! x 2 is a group homomorphism. (iii) If x 2 = 1 for all x 2 G, then G is commutative.

4 4 Exercises to Chapter Let G and H be two groups and ':G! H a group homomorphism. Show: (i) If g 2 G and n 2 N, then '(g n ) = '(g) n. (ii) If g 2 G, then ord('(g)) j ord(g). Does equality hold in general? (iii) If ' is surjective and G is commutative, then H is commutative Let G and H be two groups. (i) Let ':G! H be a map with '(g 1 g 2 ) = '(g 1 )'(g 2 ) for all g 1 ;g 2 2 G. Show that '(e G ) = e H and '(g 1 ) = '(g) 1 holds for all g 2 G. (ii) Show that a homomorphism ':G! H is injective if and only if its kernel is ker' = fe G g (i) Determine all homomorphisms S 3! Z 5. (ii) Show: For n;m 2 N >0, (Z n ; +) has a subgroup isomorphic to (Z m ; +) if and only if m divides n. (iii) Let p 2 N be prime and G;H finite groups with #G = p. If ':G! H is a homomorphism, then either '(g) = e H for all g 2 G or ' is injective Show that for k;n 2 N, the map ': Z n 7! Z n with '(a) = ka is a group automorphism of (Z n ; +) if and only if gcd(k;n) = We examine the set G of the following eight 2 2 matrices: D 0 = ; D = ; D = ; D = S 0 = ; S 1 = ; S 2 = ; S 3 = (i) The matrices D 0 ; : : : ;D 3 induce rotations of the real plane R 2 around the origin, and S 0 ; : : : ;S 3 induce reflections whose axes contain the origin. Determine the rotation angles and the reflection axes. (ii) G is a group with respect to matrix multiplication. Let U be the subset of all matrices in G that map the x-axis to itself (not necessarily pointwise). Show that U is a subgroup of G and determine the multiplication table of U. (iii) Determine all left cosets with respect to U. (iv) We consider the square with endpoints p 1 = 1 1, p2 = 1 1, p3 = 1 1, and p 1 4 = 1. Each matrix A 2 G induces a permutation 2 S4 of the four points via p (i) = A p i. For each A 2 G, find the corresponding permutation. Which subset of S 4 corresponds to U? (v) Compute the order of every element of G. Does G have a primitive element? (vi) Show that G is generated by the set fd 1 ;S 0 g, so that every element of G can be represented as a sequence of rotations by 90 and reflections about the x-axis Determine all ideals of the ring Q. : ;

5 Exercises to Chapter Show that I = f4x + 6y: x; y 2 Zg in an ideal in the ring Z Let I = f f 2 R [x]: f (5) = 0g be the set of real polynomials having 5 as a root. (i) Show that I is an ideal in R [x]. (ii) Find an isomorphism R [x]=i! R The set R = R R together with componentwise addition and multiplication is a ring. (i) Find an isomorphism of the additive groups of R and C, including a proof. (ii) Show that there is no ring isomorphism from R onto C. (Hint: R is not an integral domain.) Take R = f2k: k 2 Zg together with the usual addition and multiplication. Show: (i) (R; +) is a group, and is associative, commutative, and distributive. (ii) There is no neutral element with respect to multiplication in R Which of the following sets I are ideals in the ring R? (i) I = Z and R = Q. a b (ii) I = :a;b 2 R and R = R (iii) I = f0;3g and R = Z 6. (iv) I = f(a;0):a 2 R g and R = R R together with componentwise addition and multiplication Which of the following claims are true, which are false (give a short explanation)? (i) Z 13 has a subgroup with 5 elements. (ii) If p is prime, then the ring Z p has exactly two ideals. (iii) There is exactly one group homomorphism ': (Z 3 ; +)! (Z 5 ; +) Let R be a ring. Show that the set R = fr 2 R:9s 2 R rs = 1g of all invertible ring elements is a multiplicative group Let R be a ring (commutative, with 1) and a;b 2 R. Show that a j b if and only if b 2 hai Let R be a ring and consider R R, the set of all functions R! R. We endow R R with a ring structure in a natural way: if f ;g 2 R R, then ( f + g)(x) = f (x) + g(x); ( f g)(x) = f (x)g(x); ( f )(x) = f (x); 1(x) = 1; 0(x) = 0;

6 6 Exercises to Chapter 25 where the right-hand side operations are those of R. (The verification that R R under the above operations is a ring is trivial.) Which of the following properties of R are carried over to R R? (i) R has characteristic m 2 N, (ii) R is commutative, (iii) R is an integral domain Prove that if z 2 Z[i] is a Gaussian integer and its norm N(z) is a prime in Z, then z is irreducible in Z[i]. Verify that 1 + i;1 + 2i, and 2 3i are all irreducible in Z[i] Show that 6 and p 5 have no gcd in O Let R be an integral domain and p 2 R. Prove: (i) If p is prime, then p is irreducible. (ii) If any two nonzero elements of R have a gcd and p is irreducible, then p is prime Show that if I is an ideal in R = R 22 and I 6= f0g, then I = R. To prove this, show that ; ; ; 2 I Prove or disprove: (i) If U and V are subgroups of a group G, then so is U [V. (ii) If U and V are subgroups of a group G, then so is U \V. (iii) If F is a field, then so is F[x]. (iv) If R is a ring, then so is R[x]. (v) If R is an integral domain, then so is R[x]. (vi) Z 3 [x]=hx 2 + 1i is a field Let R be a commutative ring. Show that the following claims are true if R is an integral domain, and give counterexamples where R has zero divisors. (i) The degree formula deg( f g) = deg f + degg holds for nonzero polynomials f ;g 2 R[x]. (ii) The units of R[x] are exactly the units of R. (iii) A polynomial f 2 R[x] n f0g has at most deg f roots. Hint: Show first that x a divides f if a 2 R is a root of f (this is true in arbitrary commutative rings) Show that any finite extension field E of a field F is algebraic Let (G;) < (H;) be commutative groups. An element x 2 H is algebraic over G if it satisfies an equation of the form x n = g for some n 2 N and g 2 G, otherwise it is transcendental. H is algebraic over G if every element of H is. H is algebraically closed if every equation is solvable in H, so that for all h 2 H and n 2 N there exists an x 2 H such that x n = h. H is an algebraic closure of G if H is algebraic over G and algebraically closed. (Do not confuse these notions with the corresponding ones for field extensions.)

7 Exercises to Chapter 25 7 (i) Show that (Q ; +) is an algebraic closure of (Z; +). (ii) Let E = fz 2 C :9n 2 N z n = 1g C be the group of all complex roots of unity as in Exercise Show that (E;) is algebraically closed and that E = Q =Z. (iii) Describe the subgroup of Q =Z that is isomorphic to the subgroup T = fz 2 E:9k 2 N z 2k = 1g of E. Is T algebraically closed? (iv) Show that (Q >0 ; ) is not algebraically closed. Describe an algebraic closure G of (Q >0;) in (R >0;). Are (Q ; +) and (G;) isomorphic? (Hint: for all a;b 2 Q there exist m;n 2 Z such that ma + nb = 0.) (v) Let H 2 = h2i be the subgroup generated by 2 in Q >0 and K 2 its algebraic closure in G. Show that (K 2 ;) and (Q ; +) are isomorphic. Show that 3 is transcendental over K 2. (vi) Show that G is the direct sum of the corresponding subgroups K p for all primes p 2 N, so that every g 2 G can be uniquely written as a finite product of elements of the K p Consider the field K = Z 2 [x]=hx 3 + x + 1i. (i) What is its characteristic? (ii) What is its order? (iii) Explicitly list the elements of the field and, for two of them, give their multiplicative inverses. (iv) Give an example of an irreducible polynomial over this field of degree 2. Explain why your polynomial is irreducible For p = 7; 11, and 13, find the smallest positive integer generating the multiplicative group F p, and determine how many of the integers 1;2;3; : : : ; p 1 are generators Let p 2 N be prime, q = p k for some k 2 N >0, and F q a finite field with q elements. Show: (i) If x;y are indeterminates over F q and l 2 N, then (x + y) pl = x pl + y pl. (ii) If f 2 F q [x], then f q = f (x q ). (iii) If f 2 F q [x], then f q f = u2f q ( f u) Let p 2 N be prime, r 2 N >0, and q = p r. Under what conditions on p and r is every element of F q except 0 and 1 a generator of the multiplicative group F q? Under what conditions is every element 6= 0; 1 either a generator or the square of a generator? Prove that there is no inner product?: F 2 2 F 2 2! F Prove (5) Let q 2 R >0 or q =, and S q = fv 2 R 2 :jjvjj q = 1g. Then S 2 is the unit circle. What is S 1 and S? Describe how S q changes when q varies in the interval (0; ].

8 8 Exercises to Chapter Prove that jjajj = lim q! jjajj q for all a 2 C n Let X and Y be two discrete random variables. Prove that var(x +Y ) = var(x) + var(y ) if and only if X and Y are independent Let k 2 N and c(n) = n(logn) k for n 2 N. Prove c(2 i ) 2 O(c(n)): 0idlogne Let f ;g: N! R be eventually positive. Prove the equality O(( f + g) 2 ) = O( f 2 + g 2 ) Let f : N! R be eventually positive with lim n! f (n) = 0. Prove that 1=(1 + O( f )) = 1 + O( f ).

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