GATE CSE 2010 | Question: 4

Question

Consider the set $S = \{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S, *)$ forms

• A. A Group
• B. A Ring
• C. An integral domain
• D. A field

Comments

@srestha ma'am

is ring , field in syllabus ?

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no

ring , field out of syllabus

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The cube roots of unity can be defined as the numbers which when raised to the power of 3 gives the result as 1. In simple words, the cube root of unity is the cube root of 1 i.e.3√1.

Answer 1

Answer: A
$$\underset{\begin{array}{|c|c|c|c|} \hline \text{} & \text{1}& \omega & \omega^2 \\\hline \text{1} & \text{1}& \omega & \omega^2 \\\hline \omega & \omega & \omega^2 & \text{1}\\\hline \omega^2 & \omega^2 & \text{1} & \omega\\\hline \end{array}}{\text{Cayley Table}}$$
The structure $(S,*)$ satisfies closure property, associativity and commutativity. The structure also has an identity element $(=1)$ and an inverse for each element. So, the structure is an Abelian group.

Comments

Tuhin , ring is defined over two operations (Multiplication and addition)and they are fixed so when someone talk about ring , we can consider the binary operation operation + and · ourselves.

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use this while designing "cayley table"

1+W+W^2 =0

W^3=1

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Also, it is a cyclic group with generators $\omega,\omega^2$

Answer 2

In this question, only one binary operation is given so it cannot be a Ring or Integral domain or field.For Ring or Integral domain or for field there must be Two binary operations are required and the algebraic structure looks like (S ,+ ,⨉) .

So ,option b ,c and d are False.

 Now check the properties of groups.It satisfies all the conditions of groups.

Hence, it is a Group.

The correct answer is,(A) A Group

Answer 3

Its satisfies 1.closure property 2.associativity 3.identity element exists i.e 1 4.inverse element exist for each element So it is a group to attack these type of questions draw the composition table then it wl be easier ...

Answer 4

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