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Chapter 3: Problem 2

Show that the set of all real numbers of the form \(a+b \sqrt{2}\), where \(a\) and \(b\) are ratioral numbers, is a ficld If \(a\) and \(b\) are restncted to the integers show that this set is a ring, but 15 not a field

Short Answer

Expert verified
The set of all numbers of the form \(a+b \sqrt{2}\) with \(a,b\) rational numbers forms a field, while the same set with \(a,b\) being integers forms a ring, but not a field.

Step by step solution

01

Verify properties of Field

We consider the set \(F = \{a+b\sqrt{2} : a,b \in Q\}\) where Q is the set of all rational numbers. We need to prove it is a field by verifying below properties1. Addition and multiplication are closed in F. Let's take two elements \(x = a + b\sqrt{2}\) and \(y = c + d\sqrt{2}\). Their sum \(x + y = (a + c) + (b + d)\sqrt{2}\) and their product \(xy = (ac + 2bd) + (ad + bc)\sqrt{2}\) are both in F.2. The set F has an additive identity '0' and each element in F has an additive inverse. For any element in F, \(a + b\sqrt{2}\), the additive inverse is \(-a - b\sqrt{2}\)3. The set F has a multiplicative identity '1', and element in F except '0' has a multiplicative inverse. Except when \(a = b = 0\), which gives '0', there exists a multiplicative inverse given by \(\frac{a - b\sqrt{2}}{a^2 - 2b^2}\) which belongs to F.4. The set F is associative and commutative under addition and multiplication.
02

Verify properties of Ring

Now we consider the set \(R = \{a+b\sqrt{2} : a,b \in Z\}\) where Z is the set of all integers. We follow similar steps as in Step 1, however in proving R to be a ring we are not obligated to show the existence of multiplicative inverses. By reusing and rechecking the proofs in step 1: 1. R is closed under addition and multiplication. 2. R has additive identity '0' and each element in R has an additive inverse.3. R has a multiplicative identity '1'.4. The set R is associative and commutative under addition and multiplication, and multiplication is distributive over addition.However, unlike in Step 1, each non-zero element of R does not necessarily have a multiplicative inverse in R. Therefore, R is a ring but not a field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Field Theory
Field theory is a branch of abstract algebra that studies fields. A field is a set equipped with two operations: addition and multiplication, which follow certain rules. To determine if a set is a field, it should satisfy several key properties:
  • Closure: For any two elements from the set, the sum and product should also be in the set.
  • Additive and Multiplicative Identity: The set must have elements that act as identities for addition and multiplication. Typically, these are '0' and '1', respectively.
  • Additive and Multiplicative Inverses: Every element must have an additive inverse (such that adding it to the original number results in zero) and a multiplicative inverse (such that multiplying it by the original number results in one), except for the additive inverse of zero.
  • Associativity and Commutativity: Operations must be associative (the grouping of operands does not affect the result) and commutative (the order of operands does not affect the result).
  • Distributive Law: Multiplication must distribute over addition, meaning the expression \(a(b+c)\) equals \(ab + ac\).
In the given problem, the set \(F = \{a + b\sqrt{2} : a, b \in Q\}\) satisfies all these properties, making it a field. We confirm this by checking closure and the existence of identities and inverses for its elements.
Ring Theory
Ring theory is another important part of abstract algebra, focusing on sets that have addition and multiplication. A ring is less restrictive than a field because it does not require every element (except zero) to have a multiplicative inverse. To be classified as a ring, a set should satisfy:
  • Closure under Addition and Multiplication: The result of adding or multiplying any two elements should be an element of the set.
  • Additive Identity: There must be a 'zero' element such that adding it to any other element of the set does not change that element.
  • Additive Inverses: Each element should have an additive inverse within the set.
  • Associativity of Addition and Multiplication: The operations must be associative.
  • Distributive Property: Multiplication must distribute over addition.
For the problem provided, when considering the set \(R = \{a + b\sqrt{2} : a, b \in Z\}\), it forms a ring because it adheres to these properties while not requiring multiplicative inverses for its elements. It means every integer-only linear combination of \(1\) and \(\sqrt{2}\) follows these rules of rings but fails to guarantee a multiplicative inverse for non-integer entries, making it not a field.
Real Numbers
Real numbers are a crucial part of mathematics, especially when dealing with field and ring theories. The set of real numbers \(\mathbb{R}\) includes all rational and irrational numbers, providing a continuous sequence without gaps. Within this vast set, operations such as addition, subtraction, multiplication, and division (except by zero) are defined and satisfy field properties:
  • Archimedean Property: There are no infinitely large or small numbers, meaning there is always a real number larger or smaller than any given real number.
  • Completeness: Every non-empty subset of real numbers that is bounded has a least upper bound, making \(\mathbb{R}\) complete in terms of limits and convergence.
  • Density: Between any two real numbers, there exists another real number. This property ensures the continuity of \(\mathbb{R}\).
Understanding real numbers' relationship with abstract algebra helps in visualizing how different sets under different restrictions form fields or rings. With this comprehensive view, real numbers can be seen as an essential component of more complex algebraic structures.

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Most popular questions from this chapter

If \(L, M\) and \(N\) are vector subspaces of \(V\) show that $$ \operatorname{LO}(M+(L \cap N))=L \cap M+L \cap N $$ but it is not truc en general that $$ L \cap(M+N)=\operatorname{Ln} M+L \cap N $$
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