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Worksheet Tutorial-11
Following exercises are taken from Section 14.4 of the book
Abstract Algebra by Dummit and Foote.
1.
Determine the Galois closure of
\(\Q\left(\sqrt{1+\sqrt{2}}\right)\) over
\(\Q\text{.}\)
2.
Find a primitive generator for
\(\Q\left(\sqrt{2},\sqrt{3},\sqrt{5}\right)/\Q\text{.}\)
3.
Let
\(F\) be a field contained in
\(M_n(\Q)\text{.}\) Show that
\([F:\Q]\leq n\text{.}\)
4.
Let \(p\) be a prime number and \(F\) a field. Let \(K/F\) be a Galois extension whose Galois group is a \(p\)-group.
-
Let
\(L/K\) be a
\(p\)-extension. Show that the Galois closure of
\(L/F\) is a
\(p\)-extension of
\(F\text{.}\)
-
Give an example to show that the above result need not be true if
\(K/F\) is not Galois even if
\([K:F]\) is a power of
\(p\text{.}\)
