Given a set A in which all elements are of the form: $x+y \sqrt3$; $x,y \in \mathbb{Q}$. What structure can you define...

Question

Given a set $A$ in which all elements are of the form: $x+y \sqrt3$, $x,y \in \mathbb{Q}$.

What algebraic structure can you define with operations of addition and multiplication?

I am stuck figuring how to interpret $x+y \sqrt3$ and use it to prove the properties of algebraic structures... so any help would be sincerely appreciate.

Answer 1

For ease, I will write $r=\sqrt{3}$, so $r^2=3$.

Addition:

• It is closed: $$(x+yr)+(a+br) = (x+a)+(y+b)r$$ since $x+a$ and $y+b$ are again rational.
• $0$ is neutral element.
• $-x-yr$ is opposite to $x+yr$
• Clearly it is associative and commutative so it is Abel group for addition.

Multiplication:

• It is closed: $$(x+yr)\cdot (a+br) = (xa+3yb)+(xb+ya)r$$ since $xa+3by$ and $xb+ya$ are again rational.
• $1$ is neutral element for multiplication.
• Inverse to $x+yr\ne 0$ is $${x-ry\over x^2+3y^2}$$
• Clearly it is associative and commutative.
• And multiplication is distributive over addition...

So it is a field.

Notice that you can replace $\sqrt{3}$ with any $\sqrt{q}$ where $q$ is rational number which is not a square of rational number.