Directions. It is intended that you work these as exercises. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises either in books or in online references should not be required and is undesirable. If you make use of a reference other than class notes, you must properly cite that use.
You may not seek help from others.
If F is a field, a polynomial f(t) \in F [t] with coefficients in F determines a “polynomial function” varphi(f) from F to itself that is defined by
| (varphi(f))(a) = f(a) for a \in F . |
If A denotes the F-algebra of all functions F --> F, varphi is an F-algebra homomorphism F [t] --> A. Show the following:
varphi is injective if F is an infinite field.
varphi is not injective if F is a finite field.
varphi is not surjective if F is an infinite field.
varphi is surjective if F is a finite field.
Recall that a primitive element for a field E as an extension field of F is an element theta \in E such that E = F(theta). Find primitive elements for E over Q in the following cases:
E is the splitting field over Q of t^{12} - 1.
E = Q(SQRT{2}, SQRT{3}).
Show that t^{4} + 1 is irreducible over the field Q, but, for every prime p, not irreducible over Z/pZ. Then find the group of Q-algebra automorphisms of the field
| Q [t] / (t^{4} + 1) Q [t] . |
For each of the following irreducible polynomials of degree 3 with coefficients in Q determine the group of Q-algebra automorphisms of its splitting field:
t^{3} - 4.
t^{3} - 4 t + 2.
t^{3} - 3t - 1.
For each of the following irreducible polynomials of degree 4 with coefficients in Q determine the group of Q-algebra automorphisms of its splitting field:
t^{4} - 10 t^{2} + 1.
t^{4} -2 t^{2} - 1.
t^{4} - 4 t^{2} + 2.
t^{4} + t - 1.
t^{4} + 8 t + 12.