Let α be a complex number with α^2 = √3 − √5. Prove that Q(α)/Q is...
Let α be a complex number with α^2 = √3 − √5. Prove that Q(α)/Q is not Galois.
Homework Answers
Since Q(a)/Q is a finite (hence algebraic) extension of a field of characteristic zero, it is separable.
Here is a way to see that K=Q(a)K is not Galois. If K were Galois, then K would be invariant under any embedding K→C. However, the field K has real embeddings and complex embeddings.
In general, a Galois extension K/Q must have all of its embeddings be either real or complex, not a mix of the two. You can tell how many real embeddings K has by counting the number of real roots of the polynomial of a primitive element. In your example, the polynomial which alpha satisfies is
x8-16x4+4 has real roots and complex roots , so adjoining a root of this polynomial to Q does not give a Galois extension.
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Let α be a complex number with α^2 = √3 − √5. Prove that Q(α)/Q
is...
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH . Prove that...
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH . Prove that Q(a) equals the fixed field Q(,)". (Here with the Galois group in the usual way.) (b) Draw the diagram of all subfields of Q(13) and find primitive elements for each of them. we
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihed...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...
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.
Need help!! Please help — crypto math 1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root...
Need help!! Please help — crypto math
1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root for the prime 1223. You want to solve the discrete logarithm problem 53 (mod 1223). You know 3611 Prove it. 1 (mod 1223). Is x even or odd?
1. Determine L13(18) for p 19. 2. Let p be prime,...
Let α, β, γ ∈ ℝ designate pairwise different real numbers and understand the ℝ-vectorspace P3(ℝ)...
Let α, β, γ ∈ ℝ designate pairwise different
real numbers and understand the ℝ-vectorspace
P3(ℝ) of real polynomials of degree 2
or less as an inner product space via. = p(α)q(α)
+ p(β)q(β) + p(γ)q(γ). Now let λ ∈ C / ℝ
designate a complex number which is NOT a real number.
Question: Show that for every p, q ∈
P3(ℝ) it holds that is a real number.
(Hint: show that the number doesn't change through
complex conjugation. (NOTE:...
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1 +o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in
Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q
contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q)
is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP +
βQ|α, β ∈ Z[x]}.
(iii) For which primes p and which integers n ≥ 1 is the
polynomial xn − p...
complex analysis 4. Let J() (n2 Prove that
complex analysis
4. Let J() (n2 Prove that
Q–2: [5+2+3 Marks] Let X be a random variable giving the number of heads minus the number of tail...
Q–2: [5+2+3 Marks] Let X be a random variable giving the number of heads minus the number of tails in three tosses of a coin. a) Find the probability distribution function of the random variable X. b) Find P(−1 ≤ X ≤ 3). c) Find E(X).
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group...
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D_2p of a regular p-gon. Prove that f (x) has either all real roots or precisely one real root.
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH . Prove that Q(a) equals the fixed field Q(,)". (Here with the Galois group in the usual way.) (b) Draw the diagram of all subfields of Q(13) and find primitive elements for each of them. we
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...
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.
Need help!! Please help — crypto math
1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root for the prime 1223. You want to solve the discrete logarithm problem 53 (mod 1223). You know 3611 Prove it. 1 (mod 1223). Is x even or odd?
1. Determine L13(18) for p 19. 2. Let p be prime,...
Let α, β, γ ∈ ℝ designate pairwise different
real numbers and understand the ℝ-vectorspace
P3(ℝ) of real polynomials of degree 2
or less as an inner product space via. = p(α)q(α)
+ p(β)q(β) + p(γ)q(γ). Now let λ ∈ C / ℝ
designate a complex number which is NOT a real number.
Question: Show that for every p, q ∈
P3(ℝ) it holds that is a real number.
(Hint: show that the number doesn't change through
complex conjugation. (NOTE:...
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in
Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q
contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q)
is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP +
βQ|α, β ∈ Z[x]}.
(iii) For which primes p and which integers n ≥ 1 is the
polynomial xn − p...
complex analysis
4. Let J() (n2 Prove that
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