Exam
6320-2021-05
Summer 2021 · 5 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Determine whether the following two fields \(K\) and \(L\) are isomorphic; if they are isomorphic then construct an explicit isomorphism between them.
\[ K = \mathbb {Q}[x]/(x^2+2x+2),\quad L = \mathbb {Q}(i) \subset \mathbb {C}. \] -
Question 2
Suppose \(f(x) \in \mathbb {Q}[x]\) is a polynomial with Galois group of order \(594 = 2 \cdot 3^3 \cdot 11\). Show that \(f\) is solvable in radicals.
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Question 3
Recall that the standard representation of \(S_n\) is the subspace
\[ V_n := \{(x_1,\ldots ,x_n)\in \mathbb {C}^n,\ x_1+x_2+\cdots +x_n=0\} \]with \(S_n\) acting by permutation of the coordinates. Show that \(\wedge ^2 V_5\) is an irreducible representation of \(S_5\).
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Question 4
Show that if \(L/K\) is Galois and \(f \in K[x]\) is monic irreducible, then every irreducible factor of \(f\) in \(L[x]\) has the same degree.
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Question 5
Suppose \(\sigma \) and \(\tau \) are two elements of \(A_5\) such that, for every finite dimensional complex representation \(\rho \) of \(S_5\), \(\chi _\rho (\sigma )=\chi _\rho (\tau )\) (here \(\chi _\rho \) denotes the character of \(\rho \)). Are \(\sigma \) and \(\tau \) conjugate in \(A_5\)?