# Homework Help: Prove that its a field

1. Nov 20, 2008

### transgalactic

i got this group of numbers
http://img205.imageshack.us/img205/5291/20039746rf4.gif [Broken]

prove that its a field

??

Last edited by a moderator: May 3, 2017
2. Nov 20, 2008

### HallsofIvy

A field must be closed under addition and multiplication and every non-zero number must have a multiplicative inverse.

Is $(a+ b\sqrt{5})+ (c+ d\sqrt{5})$ in that same set?
Is $(a+ b\sqrt{5})(c+ d\sqrt{5})$ in that same set?

Is $$\frac{1}{a+b\sqrt{5}}$$ in that same set? (Hint: rationalize the denominator.)

3. Nov 21, 2008

### transgalactic

the problem is that i remember these proofs as
Q(x)+Q(y)=Q(x+y)

Q(x)*Q(y)=Q(x*y)
but here the input the root of 5 instead of sum variable

??

i dont know how to construct these equations because there is no variable input?

Last edited: Nov 21, 2008
4. Nov 21, 2008

### Staff: Mentor

Halls has shown you what you need to do. It seems pretty clear to me.

5. Nov 21, 2008

### transgalactic

i clearly remember building these equations:
Q(x)+Q(y)=Q(x+y)

Q(x)*Q(y)=Q(x*y)

but in my function there is no variable in the input

??

6. Nov 21, 2008

### sutupidmath

What is a field, you probbably already know. However, In order to show that it is a ring, alll we need to show is that it is closed under subtraction, and under multiplication. Like Halls already told you.

What you are doing there, is basically a hommomorphism, that is the perservation of the operation, but we are not dealing with mapps between two fields in here.

So let x, and y be any two elements on that set, then obviously they will have the form

$x=(a+ b\sqrt{5}); y= (c+ d\sqrt{5})$

so

$x-y=(a+ b\sqrt{5})- (c+ d\sqrt{5})=(a-c)+(b-d)\sqrt{5}$ so $$x-y \in Q(\sqrt{5})$$

now also

$$xy=(a+ b\sqrt{5})(c+ d\sqrt{5})=ac+ad\sqrt{5}+bc\sqrt{5}+bd5=(ac+bd5)+(ad+bc)\sqrt{5}[/itex] which again, for obvious reasons is in [tex] Q(\sqrt{5})$$

Now,in order for that to be a field, like Halls already stated,we need to show that every el. in that set has an inverse.

let x be in that set. so it will obviously have the form $$=x(a+ b\sqrt{5})$$ let
s suppose that there is another el, y, such that

$$xy=1 => y=\frac{1}{x}=\frac{1}{a+b\sqrt{5}}=\frac{a-b\sqrt{5}}{a^2-5b^2}$$ SO, the same question still holds, is this in the same set?

I'll give u one more hint: rewrite the last expression as

$$\frac{a}{a^2-5b^2}+\frac{-b}{a^2-5b^2}\sqrt{5}$$

7. Nov 21, 2008

### sutupidmath

you don't need to construct these, because you are not dealing with a mapping between two rings or fields, and thus you are not asked to show whether that kind of mapping is a homomorphism. because, those expressions are used when we want to show that a particular mapping say between two rings is a homomorphism.

8. Nov 21, 2008

### sutupidmath

you meant under subtraction here, right?

9. Nov 21, 2008

### transgalactic

yes if we split it:

a/(a^2-5b^2) -(b/a^2-5b^2)*5^(1/2)

is that correct?

10. Nov 21, 2008

### sutupidmath

Well, now the problem breaks down into showing whether
$$\frac{a}{a^2-5b^2}\in Q; and\frac{-b}{a^2-5b^2}\in Q ????$$

You can either prove this one too, or take it for granted. since a, b are rational nr. then by adding, subtracting, multiplying, and dividing them, we will still end up having rational numbers.

11. Nov 21, 2008

### transgalactic

hallsofIvy said to prove that that inverse of the formula is of the same set
and
a/(a^2-5b^2) -(b/a^2-5b^2)*5^(1/2) => a/(a^2-5b^2) +(-b/a^2+5b^2)*5^(1/2)

its of the same set now(it looks like the original formula)?

12. Nov 26, 2008

### transgalactic

how to prove that 0 or 1 are members
??

13. Nov 26, 2008

### HallsofIvy

If you want to write 0 or 1 as "$a+ b\sqrt{5}$", what would a and b be equal to?

14. Nov 26, 2008

### sutupidmath

$$0=0+0\sqrt{5}, 1=1+0\sqrt{5}$$ fair enough!!

P.S this is what Halls already said!...lol...

15. Dec 16, 2008

### transgalactic

how to recognize a ring proof
and field prove
i cant see the difference between them?

16. Dec 16, 2008

### NoMoreExams

17. Dec 17, 2008

### transgalactic

what is the form of a field?

what is the form of a ring?

(how each of their formula looks?)

18. Dec 17, 2008

### HallsofIvy

What are the definitions of "field" and "ring"?

19. Dec 17, 2008

### transgalactic

ring is like x=(a+b,c,g+1)
and i need to
prove that it stays the same type by addition
and multiplication by scalar
and existence of zero.

field is like a formula
f(7)=a+7*b

is this correct

20. Dec 17, 2008

### NoMoreExams

Read the wiki article I linked, what you wrote is not very mathematical in nature i.e. a field is definitely not "like a formula f(7) = a + 7*b". These are pretty advanced mathematical ideas and you should have at the very least the definition of each ingrained in your brain before you go out and try to prove things about them.

21. Dec 17, 2008

### HallsofIvy

"ring is like" is not a DEFINITION. It is very important to learn the precise words of mathematical definitions. Definitions in mathematics are "working" definitions- you use the precise words of the definitions in proofs and problems.

A "ring" is a set of objects together with two binary operations, + and *, such that:
1) The set with the operation + is an abelian group (so you need to know the definition of that!).
2) The associative law is true for *, a*(b*c)= (a*b)*c and the distributive law a*(b+c)= a*b+ a*c is true.

A "field" is a set of objects together with two binary operations, + and *, such that:
1) The set with the operation + is an abelian group.
2) The associative law is true for * and the distributive law is true.
3) There exist an "identity" for *: a*1= 1*a= a for all a in the set.
4) Every element of the set except the additive identity has a multiplicative inverse.

(3) and (4) are, of course, the difference between a ring and a field. The set of all two by two matrices over the real numbers is an example of a ring (with identity) that is not a field. Some non-zero matrices do not have inverses.

22. Dec 17, 2008

### transgalactic

regarding what you said about the field:
i remember you said that in order to prove a field
i need to prove that the sum and multiplication is of the same
structure.
and that the inverse is of the same structure
and that 0 exist

is that the tests i need to perform
?

23. Dec 17, 2008

### NoMoreExams

Halls just outlined what you need to prove. Aren't you missing a few points?

24. Dec 17, 2008

### sutupidmath

Should we also explicitly mention that it is closed under the second operation * ? i.e fo every two el. a,b from the ring R, a*b is also in the ring.

This is somehow clear since the associative law is true, but the book from which i learned these things stated it explicitly as well. So, i think it is important to state it, especially since when you want to prove that something is a subring of a ring R, then all u need to prove is that it is closed under subtraction, and multiplication.

25. Dec 17, 2008

### sutupidmath

I know for which post you are talking about, but this isn't exactly the same situation, since in that case we only needed to prove that Q is a subring. So, we already knew that it is a set of a ring. I really think you should go back and read these things, because if you are having trouble with the definition of things, i cannot immagine how you can even think about starting to prove stuff.