Q.5) Prove (1) The trivial submodule 0 is the only small submodule in Zz (2) If MN is a homomorphism and if A is small in M then (A) is small in (3) Let ABS M. If B is small in M then M is small in A

Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].

Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .

Let be as described in the proof of Theorem. Give a specific example of a positive element of .

For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.

Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].

The mapping : (Z6, +6) → (U4, ×4) on y vong ban ,0 is avouritaco at η TUOX svong bps Its euouston ai Intify ca dad ted STON n→ 3" (mod 4) (G.e) sach that yo is a homomorphism. (Here U4 = {1,3}.) (a) Write down the image of each element of Z6 under o.1 osT sula unel (b) Find Imo and Ker o. 140 1 = (512 jest, dous (II-) is aut (d) State a standard group that is isomorphic to the quotient group (c) Write down the cosets of Ker o in Z6- [8] ZG/Ker o. timil odi tadi orang of slu aham 8 - 0 58+

4. Let w = e 11, and let E = Q(w). (a) Verify that [E:Q] 10. (b) Determine |Gal(E/Q)|, the number of automorphisms of E that fix Q. (c) Find the order of y(w) w? in Gal(E/Q). (d) How many subfields K does E have (such that QC K C E)? Hint 1 The corollary to Eisenstein's theorem might help. Hint 2: Example 6, page 535 may help.

Number 6 with number 5 as help.

Page 12. (Exercise 16.22) Prove the third isomorphism theorem. At each major step in the proof, explain what is going on with a concrete example using R = Z/24Z, I = (3), and J = (6). See the book for individual steps of the proof. 13 of 15 ZOOM + »k