Is {x+y(cuberoot of 3) + z(cuberoot of 9) | x,y,z is in Q} a Subfield?

  • Thread starter lilcoley23@ho
  • Start date
In summary: S, then so is {x1x2 + 3y1z2 + 3y2z1} + {x1y2 + 3z1z2 + 3x2y1} (cuberoot 3) + {y1y2 + x1z2 + x2z1} (cuberoot 9)
  • #1
lilcoley23@ho
19
0
I'm taking an independent study class over Groups. Rings, and Fields. It's been really confusing. On one page I understand everything completely and the next page I'm completely lost. I'm looking at a problem where I has to show that {x+y(cuberoot of 3) + z(cuberoot of 9) | x,y,z is in Q} is a subring. Now I think I got that...but then I want to prove that it's a subfield. Since I prove that it's a non-empty subset and closed under addition and multiplication by showing that it's a subring, then all I further have to show is that it's a field. (Because to show something is a subfield you just have to show that it's a subset and a field.)

Then to show that the ring is a field you just have to prove it's closed under addition, subtraction, multiplication, division, and it must be cummutative...Right? So in this case I just have to prove that it's closed under division and cummutative??

I know that my denominator will look like: x^3 + 3y^3 +9z^3 - 9xyz but I really need help!


Thanks!
 
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  • #2
To show something is a field you need to show field axioms. Properties like division for example don't always make sense when talking about abstract fields.

Have you looked at showing these properties hold:

http://mathworld.wolfram.com/FieldAxioms.html
 
  • #3
So if I can show that all 5 of those properties hold for x+y(cuberoot of 3) + z(cuberoot of 9) than I can prove that x+y(cuberoot of 3) + z(cuberoot of 9) | x,y,z is in Q} is a subfield?

If that's the case can you maybe help me with the first one? Like associativity?

I would really appreciate it. I think I'm making it harder than it has to be...and where does the denominator looking like: x^3 + 3y^3 +9z^3 - 9xyz come into play?
 
  • #4
Well you want associativity for both operations. You said you have addition and multiplication so for addition, you want to show

[tex] (a + b \sqrt[3]{3} + c \sqrt[3]{9} + d + e \sqrt[3]{3} + f \sqrt[3]{9}) + g + h \sqrt[3]{3} + i \sqrt[3]{9} = a + b \sqrt[3]{3} + c \sqrt[3]{9} + (d + e \sqrt[3]{3} + f \sqrt[3]{9} + g + h \sqrt[3]{3} + i \sqrt[3]{9}) [/tex]

Can you show that just because rationals are associative under addition?
 
  • #5
I'm not sure? I know that it's proven that addition and multiplication of rational numbers are associative and cummutative, but I don't know if that's all you have to show to prove it?

Also...are you telling me that if I prove associativity, commutativity, distributity, identity, and inverses for both addition and multiplication for this equation that I have proved that it's a subfield? I thought I had to prove so much more!
 
  • #6
I just don't see that...to prove something is a subring it says you have to show that

1. S doesn't equal 0
2. S is closed under subtraction
3. S is closed under multiplication

More precisely:
1. S doesn't equal 0
2. For all a,b in S we have a-b in S and a,b in S

Then for subfield you must show that:
1. S - {0} doesn't equal 0
2. S is closed under subtraction
3. S is closed under multiplication
4. x is in S-{0} ==> x^(-1) is in S
 
  • #7
"To show something is a field you need to show field axioms" is not the only way, in fact, it is usually much easier to show that something is a subfield of something.
lilcoley23@ho, your last post contains all you need. Now that you proved that A is a subring of B. You need to do two more things, show that 1!=0, and that the set A\{0} has multiplicative inverses!

p.s What text are you using?
 
  • #8
There's a type in your last post though, in case you aren't aware of (but you probably are)
More precisely:
1. S doesn't equal 0
2. For all a,b in S we have a-b in S and a,b in S
It is For all a,b in S we have a-b in S and a.b in S
 
  • #9
So to show S was a subring of (R , + , .) I used the more precise version I talked about in my last post:

So, = 0 is in S and so, S is a non empty subset of R

Therefore, S is a subring of R.

Now condition 2:

Let say, x1+y1(cuberoot 3) + z1(cuberoot 9), x2+y2(cuberoot 3) + z2(cuberoot 9) are in S

Then, x1, x2, y1, y2, z1, z2 are in Q

So, (x1-x2), (y1-y2), (z1-z2) are in Q and (x1-x2)+(y1-y2)(cuberoot 3) + (z1-z2)(cuberoot 9) are in S

Therefore, (x1+y1(cuberoot 3) + z1(cuberoot 9)) - (x2+y2(cuberoot 3) + z2(cuberoot 9)) are in S

AND

So: (x1+y1(cuberoot 3) + z1(cuberoot 9) * (x2+y2(cuberoot 3) + z2(cuberoot 9))

==> {x1x2 + 3y1z2 + 3y2z1} + {x1y2 + 3z1z2 + 3x2y1}(cuberoot 3) + {y1y2 + x1z2 + x2z1} (cuberoot 9)

Which shomws that if x1+y1(cuberoot 3) + z1(cuberoot 9), x2+y2(cuberoot 3) + z2(cuberoot 9) are in S

Therefore: (x1+y1(cuberoot 3) + z1(cuberoot 9)) * (x2+y2(cuberoot 3) + z2(cuberoot 9)) are in S

So is that right? And if it is can you please tell me what else I have to show to show it's a subfield...

I've never seen anything with "show that 1!=0, and that the set A\{0} has multiplicative inverses"

And about my book...I'm not sure...I use mostly on-line. Really none of the problems I'm trying to solve are like any in the book!

Lilcoley23
 
  • #10
Regarding "1!=0" and " the set A\{0} has multiplicative inverses", they'are just anyway of rephrasing the subfield test, so nevermind about them.

To show it's a subfield, assume that an element exists and it doesn't equal to zero, now show that its inverse also exists.
 
  • #11
I think I haven't been quite explicit enough, sorry,

The last question about the existence of a multiplicative inverse is: is it possible to write [tex]\frac{1}{ x+y\sqrt[3]{3} + z\sqrt[3]{9} }[/tex] in the form [tex]a+b\sqrt[3]{3} + c\sqrt[3]{9}[/tex]. What values of [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] (for each x, y, and z) do we need then.
Do they belong to [tex]\mathbf{Q}[/tex].
 
  • #12
$$f$$
 
  • #13
well, so you need to show Z[cuberoot of 3] is a field.

Proof 1.
Think of the map Q[X] to Z[cuberoot of 3] that maps X to "cuberoot of 3". This is obviously a surjective ring-homomorphism. The kernel is the ideal generated by an irreducible polynomial X^3-3. Thus we have an isomorphism Q[X]/(X^3-3) onto Z[cuberoot of 3]. Since (X^3-3) is a maximal ideal, Q[X]/(X^3-3) is a field, so Z[cuberoot of 3] is also a field. ////
 
  • #14
I'm sorry,

but I dont' understand why you're just looking at Z(cuberoot 3) instead of the whole equation? Is the x+y(cuberoot of 3) part of the equation irrelevant?

Please help me understand!
 
  • #15
I denoted a ring generated by "cuberoot of 3" over Z by Z[cuberoot of 3].

Let us denote the cube root of 3 by r for the sake of notation.

Then the set in question is the same as Z[a] because a^3=3.

By the way, in general, if K is a subfield of F and a is an element of F algebraic over K, F[a]=F(a). Here, F(a) denotes the smallest subfield of F that contains K and a.

HERE IS ANOTHER PROOF (basically the same as the one I gave yesterday).

Cuberoot of 3 is a root of an irredicuble polynomial X^3-3. Any nonzero element p of the ring in question is of a form f(cuberoot of 3), where f (=:f(X)) is a polynomial over Z. Since p is not zero, X^3-3 doesn't divide f(X). (If it does, f(p)=0.) As X^3-3 is irredicible, X^3-3 and f(X) are relatively prime. Hence by the Chinese remainder theorem there are nonzero polynomials h(X) and k(X). such that h(X)(X^3-3)+k(X)f(X)=1. Substitute "cuberoot of 3" for X. Then we obtain k(cuberoot of 3)p=1. (f(cuberoot of3)=p by def.) So p has an inverse. ////
 

1. What is a subring?

A subring is a subset of a ring that is also a ring in its own right. It must contain the identity element, be closed under addition and multiplication, and have additive inverses for all elements.

2. How is a subring different from a subfield?

A subfield is a subset of a field that is also a field in its own right. In addition to the properties required for a subring, it must also have multiplicative inverses for all nonzero elements.

3. Can any subset of a ring or field be considered a subring or subfield?

No, a subset must satisfy the properties of a ring or field to be considered a subring or subfield. It must also contain the identity element and have the same operations as the original ring or field.

4. What is the significance of subrings and subfields in mathematics?

Subrings and subfields allow us to study smaller, more manageable structures within larger rings and fields. They also help us better understand the properties and relationships between elements in a ring or field.

5. How are subrings and subfields related to ideals?

Subrings and subfields are special cases of ideals, which are subsets of a ring or field that are closed under addition and multiplication by elements of the ring or field. Subrings and subfields are ideals that also contain the identity element and have the same operations as the original ring or field.

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