Let G={a+b √(2) ∈ℝ |a, b ∈ℚ} (a) Prove that G is a group under addition. (b) Prove that the nonz

Let G={a+b √(2) ∈ℝ |a, b ∈ℚ} (a) Prove that G is a group...

Key Concepts

Rationalization Techniques

Rationalization is an algebraic method used to eliminate irrational numbers from the denominator of a fraction. This technique is particularly useful in proving that the multiplicative inverse of an element with irrational components can be expressed in a form that belongs to the defined set.

Multiplicative Inverse

This is the idea that for every element (except the additive identity) in a field, there exists another element such that their product equals the multiplicative identity. Demonstrating the existence of multiplicative inverses is crucial when proving that the nonzero elements of a field form a group under multiplication.

Group Axioms

This concept involves understanding that a group is a set equipped with a binary operation that satisfies closure, associativity, has an identity element, and every element has an inverse. Verifying these properties is key when proving that a given set forms a group under the specified operation.

Field Structure

A field is an algebraic structure where two operations, addition and multiplication, are defined, and every nonzero element has a multiplicative inverse. Recognizing that the set forms a field means that, aside from being an additive group, its nonzero elements automatically form a multiplicative group.

Closure Property

A central concept in group theory, the closure property ensures that when the group operation is applied to any two elements of the set, the result is still within that set. This property must be verified for both addition and multiplication (excluding zero for multiplication) in the structure.

Transcript

00:01 Hello, should we consider the group g, which is equal to the set of a plus b times the square to two in the real numbers such that a and b are in the rational numbers.

00:12 Now to prove that g is a group under addition, we want to show that g is closed under addition, that zero is an additive identity element, and that for every element of g, there is an additive inverse in g, and that the element should follow the associative law.

00:30 We can say that if the elements, let's say, a plus b times the square root of two, and c plus d times the square root of two, if these are in g, then we have that a plus b root 2 plus c plus d root 2 in g is going to imply that a plus c plus b plus d times the square root 2 is in g.

01:00 And then since the usual addition here is commutative binary operation on the complex numbers, therefore we have the complex numbers are closed under addition, and therefore we have that the group g is closed under addition.

01:14 And we know that zero can be expressed as, well, we could write zero is equal to, i mean, zero plus zero times a square root of two.

01:22 That's still zero.

01:23 So therefore, we have that zero is in g.

01:26 And the number z can be expressed as zero plus z, which is equal to z plus zero.

01:30 So for all, a plus b square to two in the group g, we have that zero is the identity element of g.

01:44 And addition on the real numbers is associative, so therefore addition on g is also going to be associative.

01:51 And we then have proved that g is closed under addition, and therefore there exists an additive inverse in g as well, because if we let so if we let a plus b root 2, b in g, then we have negative a minus b root 2 is going to be in g.

02:16 And then this implies that a plus b root 2 plus the quantity negative a minus b root 2 is going to be equal to 0.

02:27 So we should have that every element of g is going to have an additive inverse in g.

02:33 So this then proves that g is a group under addition...

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