Proving K as a Field: Closure of Qu{i}

In summary: KFor example, 2/3 + (5/6)i = K since 2/3 + (5/6)i is a rational number and K is the set of all rational numbers.
  • #1
1LastTry
64
0

Homework Statement



Let K be the closure of Qu{i}, that is, K is the set of all numbers that can be obtained by (repeatedly)
adding and multiplying rational numbers and i, where i is the complex square root of 1.
Show that K is a Field.

Homework Equations





The Attempt at a Solution


I am having trouble starting on this problem:

What I know:

Proof the Zero vector is in the set
Proof both addition and scalar multiplication
proof additive and multiplicative inverse

^ am I missing anything?

And i am guessing I have to prove it in the form of
let Q be rational numbers
and scalars a and b in F (field)

aQ + bi = K
 
Physics news on Phys.org
  • #2
1LastTry said:

Homework Statement



Let K be the closure of Qu{i}, that is, K is the set of all numbers that can be obtained by (repeatedly)
adding and multiplying rational numbers and i, where i is the complex square root of 1.
I'm sure you mean √(-1).
1LastTry said:
Show that K is a Field.

Homework Equations





The Attempt at a Solution


I am having trouble starting on this problem:

What I know:

Proof the Zero vector is in the set
Proof both addition and scalar multiplication
proof additive and multiplicative inverse

^ am I missing anything?
Well, yes, quite a lot.
For starters, you're not dealing with vectors. Your textbook should have a definition of the axioms that define a field. You can also find them here, in the section titled "Definition and illustration" - http://en.wikipedia.org/wiki/Field_axioms.

1LastTry said:
And i am guessing I have to prove it in the form of
let Q be rational numbers
and scalars a and b in F (field)

aQ + bi = K

What "scalars" are you talking about? You need to show that a particular set, together with the operations of addition and multiplication, satisfy all of the field axioms.
 
  • #3
how do I start proofing this? I don't think i have to proof an entire list of axioms?
 
  • #4
1LastTry said:
how do I start proofing this? I don't think i have to proof an entire list of axioms?
You prove (not proof) that a set K and two operations constitute a field by showing that all of the axioms are satisfied. Again, the axioms should be listed in your book, and are also listed in the link I posted.

You can start by listing a couple of arbitrary members of the set.
 
  • #5
can you give me an example? of a member of the set?
 
  • #6
2/3 + (5/6)i
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vectors, matrices, and linear transformations. It is used to solve systems of linear equations and for various applications in fields such as physics, engineering, and computer science.

2. What are the main fields of linear algebra?

The main fields of linear algebra include vector spaces, matrices, determinants, eigenvalues and eigenvectors, and linear transformations. These concepts are used to represent and manipulate data in various applications.

3. What is a proof in linear algebra?

A proof in linear algebra is a logical argument that uses previously established theorems and definitions to demonstrate the validity of a statement or theorem. It is an essential part of mathematical reasoning and is used to verify the correctness of mathematical concepts and theories.

4. How are linear algebra and fields related?

Fields are mathematical structures that are used to define operations such as addition, subtraction, multiplication, and division. Linear algebra uses fields, such as real numbers or complex numbers, to define and perform operations on matrices and vectors. Fields are also used to define properties of linear transformations and to prove theorems in linear algebra.

5. Why is linear algebra important?

Linear algebra is important because it provides a powerful framework for representing and solving mathematical problems in various fields. It is used in many applications, including computer graphics, data analysis, and machine learning. It also provides a foundation for more advanced mathematical concepts and theories.

Similar threads

  • Calculus and Beyond Homework Help
Replies
24
Views
749
  • Calculus and Beyond Homework Help
Replies
5
Views
875
  • Calculus and Beyond Homework Help
Replies
8
Views
594
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
25
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
492
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
557
  • Calculus and Beyond Homework Help
Replies
1
Views
487
Back
Top