Algebra I. Homework 10. Due on December, 10, 2020.

1. Prove that S_{n} is generated by transpositions (i, i+ 1).

2. Prove that if n ≥3, then An is generated by 3-cycles.

3. Let L = F racC[x_{1}, . . . x_{n}] and K = L^{S}^{n} where S_{n} acts by permutations of
x_{1}, . . . x_{n}. Let σ_{i} = P

j1<j2<...jix_{j}_{1}x_{j}_{2}. . . x_{j}_{i} is the ith symmetric function.

Prove that [L : F rac(C[σ1, . . . σn])] ≤ n!. Then, using the Artin Lemma,
prove that K =F rac(C[σ_{1}, . . . σ_{n}]).

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