mathHomework代写 MAT401 Homework 9

Homework 9

mathHomework代写 These problems are due on Crowdmark by 6pm on Wednesday, March 30th. The solutions will be discussed in tutorials that week.

Read: Gallian, Chapter 32 and the Notes on Fields Extensions.

Problems:  mathHomework代写

These problems are due on Crowdmark by 6pm on Wednesday, March 30th. The solutions will be discussed in tutorials that week.

1. Find a Galois extension E/Q of degree 11.

2. Show that the polynomialis not solvable by radicals.

3. Suppose that E is the splitting fifield over Q of the polynomial

1. Q. Thus the are(i) Show that the polynomial x2 5 is irreducible in E[x].Hint: Otherwise, there would be a root β E of this polynomial and a subfifield Q(β) in E.

Now apply Galois theory.

(ii) Let K = E(5). Show that K is a Galois extension of Q with |K : Q| = 12.

(iii) Show that G = Gal(K/Q)' Z2× S3.Hint: K has subfifields E and F =Q(5) which are Galois extensions of restriction maps

rE : G −→ Gal(E/Q) and rF : G −→ Gal(F/Q)mathHomework代写

Show that the map

G −→ Gal(E/Q) × Gal(F/Q), σ 7→ (rE(σ), rF (σ))

is an isomorphism.

(iv) Let η =5 +32 K and let L = Q(η) be the subfifield it generates.

Determine

Gal(K/L) = {σ Gal(K/Q) | σ|L = idL}

and |L : Q|.

1. Suppose that G is a solvable group and that C is a subgroup. Prove that C is solvable.  mathHomework代写