18. Show that if E = F2[x]/(x4 + x +1], then E is a splitting field...
Show that the irreducible polynomial x4 - 2 over Q, has roots a, b, c in its splitting field such that the fields Q(a, b) and Q(a, c) are not isomorphic over Q (Hint: The roots are (4√2, -4√2, 4√2i, -4√2i), and the splitting field is Q(4√2, i,).)
Let KQi, 2 (a) Show that K is a splitting field of X4- 2 over Q. (b) Find a Q-basis of K c) Find an automorphism of order four of K over i (d) Determine all the automorphisms of K over Q (e) The zeros of X4-2 form -(±Vitiy2). Describe the action of the set S Aut(K) on S (f) Find all subgroups of Aut (KQ). (g) Find all intermediate field extensions of C K.
Let KQi, 2 (a) Show...
Part D,E,F,G
10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
Problem 4. Let K be a field and let f ∈ K[x]. Show that if 1+f2 has a factor of odd degree in K[x] then there is an a ∈ K such that a2 = −1.
2. a. Draw and show the crystal field splitting of the d orbitals in an octahedral complex, a tetrahedral complex, and in a square planar complex.(clearly label the orbitals) b. Explain the difference in the following crystal field splitting values (1) Co(NH3).]2+ 10200 cm"! : [CO(NH3).]* 22,900 cm (ii) [Cr(H20).]** 17,400 cm. ; [Cr(CN).]3-26,600 cm (iii) [MnF6]?- 22,200 cm-"; [ReF.]?- 27,800 cm- (iv) [Co(NH3).]2+ 10,200 cm '; [Co(NH3)4]2+ 5,900 cm c. Draw and show the filling of 6 d electrons...
Let T E L(F4, F2) be such that ker(T) = {(C1, C2, x3, x4) € F4|21 + x2 = 0 and x3 = 3x4}. Prove that T is surjective.
Q(G), where ζ7 is a primitive 7th root of 1 . Then E is the splitting field of 2. Let E x7-1 over Q (equivalently, E is the splitting field of Ф7(x) over (2). (a) Find the Gauss sums for the subfields of Gal(E/Q). (b) Exhibit the Galois correspondence between Lat(E/Q) and Sub(G), where G Gal(Q(S7)/Q). (c) Identify the fixed subfields of each subgroup (using the Gauss sums earlier com- puted).
3. Look at the inductive proof of the existence of a splitting field from our notes. If p(x) E FE is a polynomial and E is a splitting field for F, what is an upper bound on the degree of E over F? When do we have equality in this bound? (Note that this implies that splitting fields are finite extensions.)
Find the inverse of the element x^2 + x^3 + 1 in the field F2[x] / (x^4 + x^3 +1)
2.) Starting with the octahedral (ML6) crystal field splitting pattern below: A.) Draw the d-orbital splitting for a tetragonally compressed compound (ML6). This is where the ligands along the z-axis are pushed closer to the metal center, and the ligands along the x and y-axis are pulled further from the metal center (10 marks). B.) Continue this trend until the ligands along the x and y axes are completely removed, and show the d-orbital splitting for the resulting linear complex...