4. True or False. Label each of the following statements as true or false. If true,...
True or false: If N is a subgroup of a group G then the set of left cosets of N in G form a group by defining aN bN = abN for all left cosets aN and bN. Give a proof or a counterexample. (b) (5 points) True or false: If N is a subgroup of a group G then the set of left cosets of N in G form a group by defining aNbN abN for all left cosets...
Proofs are not necessary Exercise 6.8.12. Determine if the following statements are true or false. If a statement is true, prove it. If a statement is false, give a counterexample or some other proof showing it is false. Unless otherwise specified, let V and W be a finite-dimensional vector space over field F, let (v1, ..., Un} be a basis of V, let {1,...,n} be a subset of W (possibly with repeated vectors), and let 6: V W be the...
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
(3 points each) Determine whether each statement below is True or False. Give a counter-example for each false statement. (a) Every abelian group is cyclic. (b) Any two finite groups of the same order are isomorphic. (c) A permutation can be uniquely expressed as a product of transpositions. (d) Any ring with a unity must be commutative.
Solve problem 2 using the priblem 1 . Question is taken from Ring theory dealing with ideals and generating sets for ideals. Problem 1. Suppose that R (R,+ Jis a commutative ring with unity, and suppose F- (a,,. , a } is a finite nonempty subset of R. Modify your proof for Problem 5 above to show that 7n j-1 Problem 2. Consider the set Zo of integer sequences introduced in Homework Problem 6 of Investigation 16. You showed that...
for every nonzero element a n o (u)-' . Determine if the following statement is true or false, then either prove it or find a 0, then every counterexample: If A is a commutative ring with unity and char(A) nonzero element a E A has infinite order.
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that 〈a1, a2, . . . , an〉 įs an ideal of R. (If an ideal 1 = 〈a1, аг, . . . , an) for some a,...
Let R be a commutative ring with no nonzero zero divisor and elements r1,r2,.. . ,Tn where n is a positive integer and n 2. In this problem you will sketch a proof that R is a field (a) We first show that R has a multiplicative identity. Sinee the additive identity of R is, there is a nonzero a E R. Consider the elements ari, ar2, ..., arn. These are distinct. To see O. Since R conelude that0, which...
TRUE/FALSE (5 points) Answer each of the following as True or False. You don't have to justify your answer. (a) The quotient group Z12/(8) is isomorphic to Za (b) Any subgroup H of G of index 2 is normal in G. (c) For every n 2 2, the quotient group Sn/An is isomorphic to Z2. (d) If H is a normal subgroup of G, then Ha-1H for every a E H (e) The symmetric group S3 has exactly three normal...
Write a formal proof to prove the following conjecture to be true or false. If the statement is true, write a formal proof of it. If the statement is false, provide a counterexample and a slightly modified statement that is true and write a formal proof of your new statement. Conjecture: 15. (12 pts) Let h: R + RxR be the function given by h(x) = (x²,6x + 1) (a) Determine if h is an injection. If yes, prove it....