8. (a) Let R be an integral domain with the property that any strictly descending chain of ideals 11 212 213 2... is finite. Prove that R is a field.
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- (a) Draw an undirected graph G1 represented by the following adjacency matrix: 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 Does G1 contain a Hamiltonian cycle? Explain (b) Determine which pairs of graphs G1 in part (a) and G2 and G3 shown below are isomorphic and which are not, and explain your answer. If you use any invariant property in your argument, explain why this property is an invariant. Tex our answer. t 2 G2 G3 5. 7 8.Let E : y2 = x3 + 3x + 4 be an elliptic curve over F37. (a) Find all the elements of the elliptice curve group.(b) Find the order of the group.(c) Find a primitive element of this group and call it G.(d) Compute [30]G = G ⊕ G ⊕ · · · ⊕ G (addition of 30 many G’s) kindly help with full explanation. Thank you!1. Give solution, graph and if the point ( 3, y) is equidistant from ( 4, -2 ) and ( -1, 4)
- If ?(lammda) in F is a characteristic root of T then ? (lammda) is a root of the minimal polynomial of T. In particular, T only has a finite number of characteristic roots in F.4. (a) Let H be a subgroup of Sn with H not contained in An. Prove that [H: HnAn] =2. (b) Let G be a finite simple group with a subgroup H such that [G: H] = k > 2. Prove that G divides.4. Prove by direct proof that “The square of an even integer is even" 5. How many words, with or without meaning, each of 3 vowels and 3 consonants can be formed from the letters of the word EDUCATION? 6. Find whether the below graph is HAMILTONIAN GRAPH or not. Find whether the graph has a Hamiltonian Path and the Hamiltonian Circuit. A B E (F
- 9. If p(x) = 4x – 2 and q(x) = -x + 3, as a function of x, find: (a) (p o q)(x) and determine its domain and range. (b) (q • p)(x) and determine its domain and range.(a) Let H, K be subgroups of a group with identity e, such that |H| = 20 and |K|= 45. Determine |H K|, given that H n K + {e}.(a) A point D is in the interior of angle ZBAC provided that D and C are on the same side of line AB and D and B are on the same side of line AC. (b) A point D is in the interior of angle ZBAC provided that there exist points E and F on rays AB and AC, respectively, such that E * D* F. Can we use Hilbert's Axioms to prove (a) implies (b) and (b) implies (a)? Share some of your explorations. Even if you get stuck, you can share what you tried, why you tried it, and what obstacles came up.