List of top Algebra Questions

Let \( \mathbb{Q}[x] \) be the ring of all polynomials with coefficients in \( \mathbb{Q} \) under the usual polynomial addition and multiplication. Let \( T: \mathbb{Q}[x] \to \mathbb{Q}[x] \) be defined by \[ T(p(x)) = p(x^2), \quad \text{for all } p(x) \in \mathbb{Q}[x]. \] Which of the following statements is/are TRUE?

Let \( (\mathbb{Z}_n, +) \) be the group of integers modulo \(n\). Let \( G = \mathbb{Z}_{30} \oplus \mathbb{Z}_{12} \) be the external direct product of \(\mathbb{Z}_{30}\) and \(\mathbb{Z}_{12}\). Then which of the following statements is/are TRUE?

Which one of the following statements is TRUE?

Let \(x^3 - x + 1 \in \mathbb{Z}_3[x]\), where \(\mathbb{Z}_3[x]\) is the ring of all polynomials with coefficients in \(\mathbb{Z}_3\). Then the degree of the field extension

\[ \left. \mathbb{Z}_3[x] \middle/ \langle x^3 - x + 1 \rangle \right. \]

of \(\mathbb{Z}_3\) is equal to ______. (answer in integer)

Let \(\mathbb{Z}_{11}\) be the ring of integers modulo 11. Let \(\mathbb{Z}_{11}[x]\) be the ring of all polynomials with coefficients in \(\mathbb{Z}_{11}\). Which of the following statements is/are TRUE?

Let \( G \) be a group of order \( 595 \). Which one of the following is TRUE?