fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n...

Question

Question

fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n for which 2. The roots off(x) = x

Answer 1

Answer #1

lassMate ze af sik ce L 3}6 a4to let T de to af Uty Sik ce h is SIAce +fieyケーイryーイey sisth アカスッみソト HAAre my))

Answer 2

Similar Homework Help Questions

Computing another Galois Group a) Let? = eri/6 be a primitive 12th root of unity. Prove...

Computing another Galois Group a) Let? = eri/6 be a primitive 12th root of unity. Prove that is a zero of the polynomial t4 - t+ 1, and that the other zeros are 55,57,511. b) Prove that t4 – +2 + 1 is irreducible over Q and is the minimal polynomial of over Q.
6. (16 points) Let CE C be a primitive n-th root of unity. Let X =...

6. (16 points) Let CE C be a primitive n-th root of unity. Let X = 6 + 1/5. (a) (4 points) Show that Q(5) R = Q(1). (b) (4 points) Let f be the minimal polynomial of over Q. Show that Q(x) is a splitting field of f over Q. (c) (4 points) Show that Gal(Q(^)/Q) – (Z/nZ)* / (-1). (d) (4 points) Find the minimal polynomial of 2 cos(27/9) over Q.
Prove the given definition, for parts a) through c). Lemma 9.3.5 (Orthogonality Lemma). Fir N and let w-wN-e2mi/N be the natural primitive Nth root of unity in C. Fort Z/(N), we have: N-1 ktN ift-0...

Prove the given definition, for parts a) through c). Lemma 9.3.5 (Orthogonality Lemma). Fir N and let w-wN-e2mi/N be the natural primitive Nth root of unity in C. Fort Z/(N), we have: N-1 ktN ift-0 (mod N), 0 otherwise. Lukt (9.3.5) k-0 9.3.2. (Proves Lemma 9.3.5) Fix N є N, and let w-e2m/N. Let f(x)-r"-1. o510 (a) Explain why N-1 (9.3.9) (Suggestion: Try writing out the sum as 1 +z+....) (b) Explain why for any t є z/(N), fw)-0. (c)...

Prove If F is of characteristic p and p divides n, then there are fewer than n distinct nth roots...

Prove If F is of characteristic p and p divides n, then there are fewer than n distinct nth roots of unity over F: in this case the derivative is identically 0 since n=0 in F. In fact every root of x^n-1 is multiple in this case. Please write legibly, no blurry pictures and no cursive.
2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation:...

2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation: w = 1. There are exactly N distinct N-th roots of unity. Let w be a primitive root of unity, for example w = exp(2 i/N). Show the following: N, if N divides m k=0 10, otherwise N -1 N wmk 2. Fix and integer N > 2. Let f = (f(0), ..., f(N − 1)) a vector (func- tion) f : [N] →...
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...

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...

3) Given the field extensions R c F C C, such that F contains all n'th roots of unity ξ = e2mk/n,...

3) Given the field extensions R c F C C, such that F contains all n'th roots of unity ξ = e2mk/n, k-1, 2,.., n. Let 0メa E F, and let K be the splitting field of /(x) = xn-a E F[a]. T xn-a = 0, and (b) The Galois group G(K, F) is abelian hen show that: (a) K F(u) where u is any root of 3) Given the field extensions R c F C C, such that F...
Q(G), where ζ7 is a primitive 7th root of 1 . Then E is the splitting...

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).
Explain that with details thanks Topic: bilinear map and Tensor product (3) Let ơ (1, 2,...

Explain that with details thanks Topic: bilinear map and Tensor product (3) Let ơ (1, 2, ,n) E S,, be the cycle of length n. Let C, be the n x n matrix over an algebraically closed field k corresponding to σ, so Co (e) et+1 for i 1,..,n -1 and Ca(en)-e1. Show that and hence that C, is diagonalizible, similar to a diagonal matrix Dơ with diagonal entries 1,f, ξ2..-5n-1, where ξ is a primitive n-th root of unity...