Problems on Field Extensions and Galois theory

Let $F$ be a field.

Define Splitting field - Finite extension - Algebraic extension - Simple extension - Normal extension - Separable extension - Galois group - Solvable group

1. Differentiate Algebraic integer and Algebraic number.



Justify the following Statements:
1. The splitting field of a polynomial over $F$ need not be a finite extension. [TRB]

2. An algebraic extension is always a finite extension. [TRB]

3. $\mathbb{R}$ is an algebraic extension of $\mathbb{Q}$.

4. Any finite extension of a field is a simple extension. [TRB]

5. Any finite extension of a field of characteristic zero is a simple extension. [TRB]

6. Normal extension of a field is always a finite extension. [TRB]

7. Every algebraic integer is algebraic number.

8.  Every group with order less than 60 is solvable.

9. $S_n$ is solvable for $n$=
(4/6/60/30)

10. Which of the following group os not solvable?

($S_3/S_5/Z_4/Quaternion group $)