What is the Galois groups of the following polynomials?

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PROBLEM SET 15 DUE: June 16 Problem 1

What is the Galois groups of the following polynomials?

(1). x ³ − x − 1 over Q.

(2). x ³ − 10 over Q.

(3). x ³ − 10 over Q( √ 2).

(4). x ³ − 10 over Q( √

−3).

(5). (x ² − 2)(x ² − 3)(x ² − 5)(x ² − 7) over Q.

(6). x ⁿ − t where t is transcendental over the complex numbers C and n is a positive integer, over C(t).

Problem 2

Let k be a field of characteristic 6= 2. Let c ∈ k, c / ∈ k ² . Let F = k( √ c).

Let α = a + b √

c with a, b ∈ k and not both a, b = 0. Let E = F ( √ α).

Prove that the following conditions are equivalent:

(1). E is Galois over k.

(2). E = F ( √

α ⁰ ), where α ⁰ = a − b √ c.

(3). Either αα ⁰ ∈ k ² or cαα ⁰ ∈ k ² .

Show that when these conditions are satisfied then E is cyclic over k of degree 4 if and only if cαα ⁰ ∈ k ² .

Problem 3

Show that the regular pentagon is ruler and compass constructible.

Problem 4(cyclotomic polynomials) (1). Let Φ d (x) = Q

periodζ=d (x − ζ), where the product is taken over all n − th roots of unity of period d. Show that

Φ n (x) = x ⁿ − 1 Q

d|n,d<n Φ d (x) . And compute Φ _n (x) for 1 ≤ n ≤ 12.

(2). Φ n (x) is called the n-th cyclotomic polynomial. Show that if

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p is a prime number, then

Φ p (x) = x ^p−1 + x ^p−2 + .... + 1.

and for an integer r ≥ 1,

Φ p

(x) = Φ p (x ^p

).

(3). Let n = p ^r ₁

....p ^r _s

be a positive integer with its prime factorization.

Then

Φ n (x) = Φ p

....p

(x ^p

^....p

).

(4). If n is odd > 1, then Φ 2n (x) = Φ n (−x).

(5). If p is a prime number not dividing n, then Φ pn (x) = Φ _n (x ^p )

Φ n (x) . On the other hand, if p|n, then Φ pn = Φ n (x ^p ).

(6). We have

Φ n (x) = Y

d|n

(x

− 1) ^µ(d) . As usual, µ is the Mobius function.

Problem 5

Show that P

x∈F

x ^m = 0 for q not dividing m, where F q is a finite field and the summation is taken over all elements x ∈ F q . While P

x∈F

x ^m = −1 for q dividing m.

Problem 6

(1). Let K 1 , ...., K n be Galois extensions of k with Galois groups G 1 , ...., G n . Assume that K i+1 ∩ (K 1 ....K i ) = k for each i = 1, 2, ...., n − 1.

Then the Galois group of K 1 ....K n is isomorphic to the product G 1 ×....×G n

in a natural way.

(2). Let p 1 < p 2 < .... < p n < .... be a sequence of prime numbers, and K ₁ = Q( √

p ₁ ), K _i+1 = K _i ( √

p _i+1 ). Compute the Galois group of K ₁ ....K _n

over Q.