3. Consider the field Q(VB, i). (a) Is this a splitting field for some polynomial in...
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,).)
Cs Saaet be the splitting field of a polynomial f of degree 5 over Q. Prove that E has no subfields F with [F : =7. Cs Saaet be the splitting field of a polynomial f of degree 5 over Q. Prove that E has no subfields F with [F : =7.
Problem 3: Determine the splitting field of the polynomial (2 -2)(2-3)(2 -4) over Q. Find its degree over Q. Verify if all points of the splitting field are constructible. Problem 3: Determine the splitting field of the polynomial (2 -2)(2-3)(2 -4) over Q. Find its degree over Q. Verify if all points of the splitting field are constructible.
5. Let F be a field, and let p(x) ∈ F [x] be a separable, irreducible polynomial of degree 3. Let K be the splitting field of p(x), and denote the roots of p(x) in K by α1, α2, α3. a) (10’) If char(F ) does not equal 2, 3, prove that K = F (α1 − α2).
8. For each of the equations listed below, determine the Galois group over Q of the splitting field of the equation. List all of the subgroups of the Galois group. List all of the subfields of the splitting field of the equation, and draw a diagram illustrating the Galois correspondence between subgroups and subfields for each example. a. 2 1) (z2-2) b.(-3) +1) (Note: You must prove by explicit calculation that /3 is not contained in QlV2.) 3 8. For...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
just part c and d 3. Consider the group Z45 (a) Give an example of a subnormal series which is not a composition series. Explain why your example works. (b) Find a refinement of the series you gave above which is a composition series. That is, show how to extend your series into a composition series. (c) Find all other composition series and briefly explain how you know you have them all. (d) If p(x) is a polynomial whose splitting...
B. Let p and q be distinct positive prime numbers. Set a p+ (a) Find a monic polynomial f(x) EQlr of degree 4 such that f(a) 0. (b) Explain why part (a) shows that (Q(a):QS4 (c) Note: In order to be sure that IQ(α) : Q-4, we would need to know that f is irreducible. (Do not attempt it, though). Is it enough to show that f(x) has no rational roots? (d) Show V pg E Q(α). Does it follow...
i need a, b, and c Note: a lowercase letter indicates a q The PlLb address in your Lab Narrative questions is to prompt / reminough answetion ok. The pul to write in your to a few questions is Thoughtful and thorough answers ferred over quick answers o every question rticle in a Magnetie Field Demo Description of e/m apparatus: air ch of helium gas is inserted. We it glows in The heart of this instrument is a evacuated and...
i need help on question 3 to 22 please. Midterm ex review. MATH 101 Use the following information to answer the next four exercises. The midterm grades on a chemistry exam, graded on a scale of 0 to 100, were: 62, 64, 65, 65, 68, 70, 72, 72, 74, 75, 75, 75, 76,78, 78, 81, 82, 83, 84, 85, 87, 88, 92, 95, 98, 98, 100, 100,740 1. Do you see any outliers in this data? If so, how would...