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.
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)
(4/6/60/30)
10. Which of the following group os not solvable?
($S_3/S_5/Z_4/Quaternion group $)
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