Problem 5. Let E1 = Q(2,7 ), E2= (2,), 1 = 22 + 77, and 2 = 22 + 3()
(i) Determine [Ei : Q] for i = 1, 2.
(ii) Determine a basis of Ei over Q for i = 1, 2.
(iii) Determine the minimal polynomial of
i over Q for i = 1, 2.
(iv) Determine if each of the extensions E1 / Q and
E2 / Q is Galois.
(i). Minimal polynomial of over Q is and so . Similarly minimal polynomial of over is and so . Hence, .
Minimal polynomial of over is and so . Hence, .
(ii). A basis of over is and a basis of over is . Hence, a basis of over is .
A basis of over is . Hence a basis of over is .
(iii). Let and . Now, . And . So,. Since is a root of , is a root of and hence the minimal polynomial of is the above 4 degree polynomial.
Take . Then, and . Similarly as above, since is a root of minimal polynomial of is .
(iv). Note that, is the splitting field of the polynomial over . Since a splitting field is always a normal extension, we get that is a finite normal extension. Also since characteristics of is 0, any field extension of is separable. Hence, is a finite, normal, separable extension and so it is a Galois extension.
Now note that, is an irreducible polynomial over and contains a root ( namely, ) of this polynomial. Now if this was a normal extension, then have to contain all the roots the above irreducible polynomial. But is a root of the polynomial ( is a complex cube root of 1) and we can clearly see that does not contain . Hence, is not a normal extension and so it's not a Galois extension.
Abstract Algebra: Let . It has been shown already that K is the splitting field over , and the following isomorphisms are of onto a subfield as extensions of the automorphism , and also the elements of : ; ; ; . We also proved previously that is separable over . Based on all of those outcomes, find all subgroups of and their corresponding fixed fields as the intermediate fields between and , and complete the subgroup and subfield diagrams...
which of the following procedures will yield the same estimate of 1 as in multiple regression Y=0 + 1122+U ? A. Run Y on 1, predict residual 1; run Y on 2, predict residual e2; run e1 on e2 B. Run X2 on X1 predict residual e; run e on Y C.Run Y on X1 predict residual e1; run X2 on X predict residual e2; run e1 on e2 E. none of the above We were unable to transcribe this...
Part 1: For each of the following structures, indicate the integration expected for the signal associated with the indicated hydrogen(s). a) i) ii) iii) iv) b) i) ii) iii) c) i) ii) iii) d) i) ii) iii) iv) v) vi) e) i) ii) iii) iv) f) i) ii) iii) iv) v) vi) Part 2: For each of the following structures, indicate the coupling (a.k.a, splitting) pattern expected for the signal associated with the indicated hydrogen(s) by placing the appropriate letter(s)...
Let be a prime and let be the set of rational numbers whose denominator (when written in lowest terms) is not divisible by . i) Show, with the usual operations of addition and multiplication, that is a subring of . ii) Show that is a subring of . iii) Is a field? Explain. iv) What is where is the set of all fractions with denominator a power of We were unable to transcribe this imageWe were unable to transcribe this...
Let be an arbitrary function and A X. i) Show that A ii) Give an example to show that in general A = . iii) Show that, if is injective, then A = iv) Show that, if X and Y are modules; is a homomorphism of modules and A is a submodule of X such that ker, then we also have A = We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe...
I am having trouble differing between Sn1, Sn2, E1, and E2 reactions. Please help me understand how to approach these problems. I've attempted to answer them, but I am not confident in my answers. Please show me how to get to the correct answer. 1. [is it A?] 2. [Is it C?] 3. [is it 3 and 4?] A) I B) II C) III D) IV E) I and II F) III and IV Predict the mechanism (S\2, E2, SN1,...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) = –2 + x. Let E = (e1,e2) and B = (P1, P2). 2 a) Show that B is a basis for P1(R). 4 b) Let ce : R → R3 be the change of coordinates from E to ß. Find the matrix representation of C. Leave your answer as a single simplified matrix. 6 c) Let (:,:) be an inner product on P1(R). Suppose...
Let be i.i.d. . Define the sample mean and the sample variance by and . (i) Find the distribution of and for i = 1, ... , n. (ii) Show that and are independent for i = 1, ... , n. (iii) Hence, or otherwise, show that and are independent. 7l N (μ, σ2) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
3. Let ,..., be independent random sample from N(), where is unknown. (i) Find a sufficient statistic of . (ii) Find the MLE of . (iii) Find a pivotal quantity and use it to construct a 100(1–)% confidence interval for . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
IV. Let (10 -3 2 A= 0 1 -54 3 -2 1 -2 (a) Find a basis for the null-space of A. (b) Find a basis for the column-space of A. We were unable to transcribe this image