We have to determine whether the set of all integers of the form m+(n(sqrt(3))) is a ring w.r.t to addition and multiplication.
Is there any identity element e which belongs to this set, under addition such that x + e = x = e+x for all x belongs to above set?
Since e should also be of the form m+(n(sqrt(3))), and e should be 0, which is, e = 0+(0*sqrt(3))), can we take this belongs to the set since both m=n=0?
We have to determine whether the set of all real numbers of the form m+(n(sqrt(3))) , where m,n are integers, is a ring w.r.t to addition and multiplication*
looks like e=0+0√3 works for addition
for multiplication, I get
(a+b√3) = (-m+n√3)/(3n^2-m^2)
So, it looks like there is no multiplicative identity if 3n^2 = m^2
Luckily, there are no integers where this is true.
so, for example, (2+5√3)*(-2/71 + 5/71 √3) = 1
To determine if the set of all integers of the form m + n(sqrt(3)) is a ring with respect to addition and multiplication, we need to check several properties.
First, let's verify if there exists an identity element under addition (denoted as e) such that for all x in the given set, x + e = x = e + x.
To find the identity element e, we need e to satisfy the following equation: x + e = x.
In the given set, an element x can be represented as m + n(sqrt(3)). So we have:
(m + n(sqrt(3))) + e = m + n(sqrt(3)).
Expanding the left side of the equation:
m + n(sqrt(3)) + e = m + n(sqrt(3)).
Since the coefficients of sqrt(3) must be equal, we have:
m + e + n(sqrt(3)) = m + n(sqrt(3)).
In order for this equation to hold for all m and n, the constant term should be equal on both sides:
m + e = m.
This implies that e = 0.
So the identity element, e, is indeed 0.
Therefore, the set of all integers of the form m + n(sqrt(3)) is a ring with respect to addition since it has an identity element (0) under addition.
Note: Keep in mind that we have only verified the existence of an identity element under addition. To determine if this set is a ring, we also need to check other ring properties, such as closure, associativity, commutativity, and the existence of inverses under addition and multiplication.