List of top Algebra Questions asked in GATE MA

The number of 5-Sylow subgroups in the symmetric group \( S_5 \) of degree 5 is \( \underline{\hspace{1cm}}\).

Let \( I \) be the ideal generated by \( x^2 + x + 1 \) in the polynomial ring \( R = \mathbb{Z}_3[x] \), where \( \mathbb{Z}_3 \) denotes the ring of integers modulo 3. Then the number of units in the quotient ring \( R/I \) is \(\underline{\hspace{1cm}} \).

Let \( G \) be a group of order \( 5^4 \) with center having \( 5^2 \) elements. Then the number of conjugacy classes in \( G \) is \(\underline{\hspace{1cm}}\) .
Let \( F \) be a finite field and \( F^{\times} \) be the group of all nonzero elements of \( F \) under multiplication. If \( F^{\times} \) has a subgroup of order 17, then the smallest possible order of the field \( F \) is \(\underline{\hspace{1cm}}\) .
Let \( \mathbb{Z} \) denote the ring of integers. Consider the subring \[ R = \{ a + b\sqrt{-17} : a, b \in \mathbb{Z} \} \] of the field \( \mathbb{C} \) of complex numbers. Consider the following statements: P: \( 2 + \sqrt{-17} \) is an irreducible element.
Q: \( 2 + \sqrt{-17} \) is a prime element. Then: