Question

True or False, explain:

1. Any polynomial f in Q[x] with deg(f)=3 and no roots in Q is irreducible.

2. Any polynomial f in Q[x] with deg(f)-4 and no roots in Q is irreducible.

3. Z^x₄₀ is isomorphic to Z^x₅ x Z^x₈

4. If G is a finite group and H<G, then [G:H] = |G||H|

5. If [G:H]=2, then H is normal in G.

6. If G is a finite group and G<S₂₈, then there is a subgroup of G of order 2401=7⁴

7. If |G|=10, then G is isomorphic to Z₁₉.

8. If F subset of K is a degree 5 field extension, any element b in K is the root of some polynomial p(x) in F[x]

9. If F subset of K is a degree 5 field extension, viewing K as a vector space over F, Aut(K, F) consists of all F-linear transformationf of K.

Answer 1