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Exercises 11.4 Exercises
1.
Determine the Galois groups of each of the following polynomials in \({\mathbb Q}[x]\text{;}\) hence, determine the solvability by radicals of each of the polynomials.
\(\displaystyle x^5 - 12 x^2 + 2\)
\(\displaystyle x^5 - 4 x^4 + 2 x + 2\)
\(\displaystyle x^3 - 5\)
\(\displaystyle x^4 - x^2 - 6\)
\(\displaystyle x^5 + 1\)
\(\displaystyle (x^2 - 2)(x^2 + 2)\)
\(\displaystyle x^8 - 1\)
\(\displaystyle x^8 + 1\)
\(\displaystyle x^4 - 3 x^2 -10\)
2. Let
\(F \subset E\text{.}\) If
\(f(x)\) is solvable over
\(F\text{,}\) show that
\(f(x)\) is also solvable over
\(E\text{.}\)
3. Construct a polynomial
\(f(x)\) in
\({\mathbb Q}[x]\) of degree
\(7\) that is not solvable by radicals.
4. Let
\(p\) be prime. Prove that there exists a polynomial
\(f(x) \in{\mathbb Q}[x]\) of degree
\(p\) with Galois group isomorphic to
\(S_p\text{.}\) Conclude that for each prime
\(p\) with
\(p \geq 5\) there exists a polynomial of degree
\(p\) that is not solvable by radicals.
5. Let
\(K\) be the splitting field of
\(x^3 + x^2 + 1 \in {\mathbb Z}_2[x]\text{.}\) Prove or disprove that
\(K\) is an extension by radicals.
