Which of the following statements are true? Give reasons for your answ

Question

Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid justification of your answers.

(i) If G is a finite abelian group and p is a prime factor of o(G), then the number of Sylow p-subgroups of G is a prime.

ii) The minimum polynomial of 1/3 5 over Q is x^1/3 .

iii) $\mathbb{Z}_{mn}\neq \mathbb{Z}_{m}\mathbb{Z}_{n}m,n\in \mathbb{Z}$

iv) If G is a finite group and m o(G), $m\epsilon n$ then G has a subgroup of order m.

v) If $\beta _{1}$ and $\beta _{2}$ are two 15^th roots of unity, then $\mathbb{Q} (\beta _{1}) =\mathbb{Q} (\beta _{2})$

vi) There exists an extension field of $\mathbb{Z}_{3}$ of order 25.

vii) Every group of order 18 has a normal subgroup of order 2.

viii) If I and J are ideals of a ring R, then IJ = $I\cap J$

ix) If f : R S → is a ring homomorphism and I is an ideal of R, then f (I) is an ideal of S.

x) Every prime ideal of an integral domain is a maximal ideal.

29 Nov 2023

Answer :

Word Count : 607

Let's go through each statement one by one and analyze whether it is true or false, providing reasons for each answer.

(i) If G is a finite abelian group and p is a prime factor of o(G), then the number of Sylow p-subgroups of G is a prime.

Answer: True.

Reason: In a finite abelian group, every subgroup is normal, and the number of Sylow p-subgroups is a divisor of the order of the group. Since p is a prime factor of the order of G, the number of Sylow p-subgroups must be a prime.

(ii) The minimum polynomial of 1/3 5 over Q is x^(1/3).

Answer: False.

Reason: The minimum polynomial of 1/3^(5) over Q is x^(5) - 3. The ____ ____ ______ _________ ______ _________ _________ ________ _______ _________.
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