Proof of closure under addition and multiplication in a field

In summary: But then I realized that depending on what you mean by "a field", you might not be able to show that a+b and ab are in the same field. For example, if you mean the field of real numbers, then a+b and ab are in the same field, because the real numbers are a subset of the real numbers. But if you mean the field of complex numbers, then a+b and ab are not in the same field, because the complex numbers are not a subset of the real numbers.
  • #1
karnten07
213
0

Homework Statement




Does anyone know a generic way of showing that a field is closed under multiplication and addition? Please, thanks

Homework Equations





The Attempt at a Solution



Just need to prove that a+b and ab are in the field that each element a and b are from. Any ideas??
 
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  • #2
There's not really a generic way to do it. It all depends on what set you're claiming is a field. Incidentally, calling it a field before you show it's closed under the operations is bad form.
 
  • #3
Mystic998 said:
There's not really a generic way to do it. It all depends on what set you're claiming is a field. Incidentally, calling it a field before you show it's closed under the operations is bad form.

The field I am trying to show is a field is of the form Q[tex]\sqrt{}[/tex]d where Q is the set of rational numbers and d is in the set of complex numbers, C.
 
  • #4
karnten07 said:
Does anyone know a generic way of showing that a field is closed under multiplication and addition? Please, thanks
Yes -- the definition of "field" mandates that it is closed under multiplication and addition.

I suspect you meant to ask something else -- judging by your wording, did you mean to ask about showing whether a subset of a field (with the induced arithmetic operations) is a subfield?
 
  • #5
Okay, so what you want to do is take arbitrary elements of the field, namely elements of the form [itex]a + b\sqrt{d}[/itex] and [itex]c + e\sqrt{d}[/itex], with [itex]a, b, c, e \in \mathbb{Q}[/itex], and show that you get something of that form back when you multiply or add them.
 
  • #6
As Mystic988 said in the first post, how you show a set if closed under operations (and so is a field) depends on how the set and operations are defined. How you would show that "a field" is of the form Q([itex]\sqrt{d}[/itex]) where d is a complex number depends upon exactly how "a field" is defined!

Exactly how is your field defined? Mystic988's suggestion is assuming you already know d and want to show that the set of numbers of the form a+ b[itex]\sqrt{d}[/itex] is a field. That is quite correct if d is an integer but if, for example, d is a trancendental number, it is not at all correct.

Again, exactly what is the problem you are working on?
 
  • #7
HallsofIvy said:
As Mystic988 said in the first post, how you show a set if closed under operations (and so is a field) depends on how the set and operations are defined. How you would show that "a field" is of the form Q([itex]\sqrt{d}[/itex]) where d is a complex number depends upon exactly how "a field" is defined!

Exactly how is your field defined? Mystic988's suggestion is assuming you already know d and want to show that the set of numbers of the form a+ b[itex]\sqrt{d}[/itex] is a field. That is quite correct if d is an integer but if, for example, d is a trancendental number, it is not at all correct.

Again, exactly what is the problem you are working on?

It's okay, i did the assignment now, i apologise for my poor wording of the question. But i understand how to do the question now, thanks guys
 
  • #8
Oh, yeah, I was assuming we were talking about quadratic rationals (I think that's the right name. My brain is not functioning fully at this very moment), not more exotic field extensions. Sorry about that.
 
  • #9
I started to answer the question in exactly the same way!
 

1. What does "closure under addition and multiplication" mean in a field?

Closure under addition and multiplication in a field means that when two elements in the field are added or multiplied, the result is also an element of the field.

2. Why is closure under addition and multiplication important in a field?

Closure under addition and multiplication is important in a field because it ensures that the field is a closed and self-contained system, allowing for consistent mathematical operations and properties.

3. How is closure under addition and multiplication proven in a field?

To prove closure under addition and multiplication in a field, it must be shown that the result of adding or multiplying any two elements in the field is also an element of the field. This can be done through mathematical proofs or by showing that the field satisfies the axioms of a field.

4. Is every set with addition and multiplication a field?

No, not every set with addition and multiplication is a field. A field must also satisfy additional axioms, such as the existence of multiplicative inverses and the commutative and distributive properties, in addition to closure under addition and multiplication.

5. Can a field have more than two operations and still be "closed"?

Yes, a field can have more than two operations and still be "closed". In fact, a field must have two operations, addition and multiplication, as well as their inverses, to satisfy the axioms of a field. However, some fields may also have other operations, such as exponentiation or logarithms, that are defined in terms of addition and multiplication.

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