prove The Chinese remainder theorem by the steps in the picture Proof of CRT (for two modli)We must prove that f: Zm,m.Zmx Z giuen byis abijection.ProofO Fist check that f is well- debinad, ine. that f is indeperdentof the choice of representative x for the class [x)2 Show that f is one-to-oneO Show that ļ is ontoThis is trünial if f is one-to-one, as !2mm= m,m, ! Zm, !:1 Zma !.

Given: The function, f:ℤm1m2→ℤm1×ℤm2 defined by fxm1m2=xm1,xm2. To prove: The given function is a…

Answered: Proof of CRT ( for two modi) We must…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 24E

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Let S be a nonempty subset of an order field F. Write definitions for lower bound of S and greatest lower bound of S. Prove that if F is a complete ordered field and the nonempty subset S has a lower bound in F, then S has a greatest lower bound in F.
Let p be prime and G the multiplicative group of units Up=[a]Zp[a][0]. Use Lagranges Theorem in G to prove Fermats Little Theorem in the form [a]p=[a] for any aZ. (compare with Exercise 54 in section 2.5) Let p be a prime integer. Prove Fermats Little Theorem: For any positive integer a,apa(modp). (Hint: Use induction on a, with p held fixed.)
49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove that if is a prime integer, then is a factor of for . (From Exercise of Section, it is known that is an integer.) b. Use induction on to prove that if is a prime integer, then is a factor of .
Prove that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e denotes the multiplicative identity in F. (Hint: See Theorem 5.34 and its proof.) Theorem 5.34: Well-Ordered D+ If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then e is the least element of D+ and D+=nen+.
34. Let be the set of eight elements with identity element and noncommutative multiplication given by for all in (The circular order of multiplication is indicated by the diagram in Figure .) Given that is a group of order , write out the multiplication table for . This group is known as the quaternion group. (Sec. Sec. Sec. Sec. Sec. Sec. Sec. ) Sec. 22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. 32. Find the centralizer for each element in each of the following groups. a. The quaternion group in Exercise 34 of section 3.1 Sec. 2. Let be the quaternion group. List all cyclic subgroups of . Sec. 11. The following set of matrices , , , , , , forms a group with respect to matrix multiplication. Find an isomorphism from to the quaternion group. Sec. 8. Let be the quaternion group of units . Sec. 23. Find all subgroups of the quaternion group. Sec. 40. Find the commutator subgroup of each of the following groups. a. The quaternion group . Sec. 3. The quaternion group ; . 11. Find all homomorphic images of the quaternion group. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined by

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