Texte intégral
Algebra I. Homework 4. Due on October 8, 2020.
1. Prove or disprove: a subring of a Noetherian ring is Noetherian.
2. Letpbe a prime number. Show that the polynomial1 +x+. . .+xp−1 inQ[x]
is irreducible.
3. Prove that f(x) =x4+ 4x+ 1∈Q[x]is irreducible.
4. Let R =C[x, y] and let I be the ideal of R generated by two elements(x, y).
Prove or disprove: I is a free R-module.
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