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The polynomial 23 - 2 + 1 has no roots in Zg, so it is irreducible...
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,).)
omialS 1. (a)Tell why the polynomial +22 must be irreducible over the rational numbers. (b) What does this mean about the field EQ/r3 2 +3)? (c) What is a basis for this field, considered as a field extensions of its field of constants, isomorphic to Q? (d) What is the degree of the extension E:Q? (e) Let y denote the equivalence class of z in E. Add and multiply these two elements of E: 1 + 2y + y2,y-Zy2 and...
8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the denomina- tor, d runs through all divisors of n less than n). . A. Calculate Ф2(x), Ф4(x) and Ф8(z): . B. n(x) is the minimal polynomial for the primitive n-th root of unity over Q. Let f(x) = "8-1 E Q[a] and ω...
The polynomial of degree 5, P(2) has leading coefficient 1, has roots of multiplicity 2 at I = 1 and I = 0, and a root of multiplicity 1 at I = - 3 Find a possible formula for P(x). P(x) = Question Help: Video Submit Question
Please answer A, B, and C in full 2. Let f() € F[2] be a separable polynomial with roots {u1, ..., Un} contained in some splitting field K of f(x) over F. Define A= || (ui-u) = (ui - U2) (u - u3) ...(ui-un)(uz - u3) ..(un-1 - Un) EK. (a) (15 points) Consider GalpK < Sn by looking at its action on the set of roots for f(x). Show that if Te Galo K is a transposition then (A)...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...