1. ## order prpoperties

Let k:= {s+t(sqrt)2 : s, t exist in Q (rational numbers). Show that K satisfies the following:
1. If x,y exist in K then x+y exists in K and xy exists in K
2. If x is not 0 and x is in K then 1/x is in K

(Thus the set K is a subfield of R. With the order inherited from R the set K is an ordered field that lies between Q and R)
Also im not sure what this little tidbit means either

Thank you!

2. Originally Posted by mremwo
(Thus the set K is a subfield of R. With the order inherited from R the set K is an ordered field that lies between Q and R)
Also im not sure what this little tidbit means either
$\displaystyle K$ is an ordered field with the order inherited from $\displaystyle \mathbb{R}$ because we verify:

$\displaystyle (i)\;x\leq y\Rightarrow x+z\leq y+z\quad ( \forall{z}\in K)$

$\displaystyle (ii)\;x\leq y\Rightarrow xz\leq yz\quad (\forall{z}\in K\wedge z\geq 0)$

Fernando Revilla

3. So, $\displaystyle K=\{s+t\sqrt{2}\mid s,t\in\mathbb{Q}\}$. (One usually says "s is in Q," not "s exists in Q.") Suppose $\displaystyle s+t\sqrt{2}\in K$. Then $\displaystyle \displaystyle\frac{1}{s+t\sqrt{2}}=\frac{s-t\sqrt{2}}{(s+t\sqrt{2})(s-t\sqrt{2})}= \frac{s-t\sqrt{2}}{s^2-2t^2}=\frac{s}{s^2-2t^2}-\frac{t}{s^2-2t^2}\sqrt{2}\in K$. Checking point 1 is easier.

(Thus the set K is a subfield of R. With the order inherited from R the set K is an ordered field that lies between Q and R)
Also im not sure what this little tidbit means either
$\displaystyle K$ is a subfield of $\displaystyle \mathbb{R}$: every element of K is a real number and $\displaystyle K$ satisfies the axioms of a field.

With the order inherited from $\displaystyle \mathbb{R}$ the set $\displaystyle K$ is an ordered field: with the regular order, $\displaystyle K$ satisfies the axioms of an ordered field. Fernando showed two axioms of an ordered field that come on top of axioms of a field.

...lies between $\displaystyle \mathbb{Q}$ and $\displaystyle \mathbb{R}$: $\displaystyle \mathbb{Q}\subset K\subset\mathbb{R}$.

4. Im still not understanding!

5. What exactly are you not understanding?

You need to prove
1. If x,y exist in K then x+y exists in K
What should the first phrase of the proof be?

6. i dont get how you know if it is in K
just if it is in the form of "s+tsqrt2"?

Also, would i make y (in the original problem it is xsub1) ssub1 + tsub1sqrt2 ? and then say that s+tsqrt2,ssub1 + tsub1sqrt2 is in k?

7. Originally Posted by mremwo
i dont get how you know if it is in K
just if it is in the form of "s+tsqrt2"?
What is "it" in your question? The notation $\displaystyle \{s+t\sqrt{2}\mid s,t\in\mathbb{Q}\}$ is used to denote a set that has all possible elements of the form $\displaystyle s+t\sqrt{2}$ for $\displaystyle s,t\in\mathbb{Q}$ and only such element. So, something of the form $\displaystyle u+v\sqrt{2}$ is in this set, called K here, and every element of K has this form.

Also, would i make y (in the original problem it is xsub1) ssub1 + tsub1sqrt2 ?
First, I am not sure what you mean by the "original problem." For me, the original problem is the first post of this thread, and it does not have $\displaystyle x_1$. Yes, if $\displaystyle y\in K$, then y has the form $\displaystyle s_1+t_1\sqrt{2}$ for some rational $\displaystyle s_1,t_1$.

and then say that s+tsqrt2,ssub1 + tsub1sqrt2 is in k?
Is is not clear to me what s and t are, how they were introduced. In any case, if $\displaystyle s,s_1,t,t_1$ are rational, then yes, $\displaystyle s+t\sqrt{2}\in K$ and $\displaystyle s_1+t_1\sqrt{2}\in K$, by definition of K. I am not sure how this advanced the proof, though.

To answer my question from the previous post:
What should the first phrase of the proof be?
It should be: "Suppose some arbitrary x, y are in K." In general, a proof of a statement of the form "If A, then B" starts with "Suppose A." Then, since $\displaystyle x,y\in K$, there exists $\displaystyle s,t,s_1,t_1\in\mathbb{Q}$ such that $\displaystyle x=s+t\sqrt{2}$ and $\displaystyle y=s_1+t_1\sqrt{2}$. Note that I introduced $\displaystyle s_1,t_1$ by saying "there exist". Also, the point was not to say that $\displaystyle s_1+t_1\sqrt{2}\in K$, which is a triviality; the point was to say that $\displaystyle y=s_1+t_1\sqrt{2}$. After that, you can see what x + y looks like.

Hint: it is customary to write x1 or x_1 instead of xsub1. On the other hand, it is better to write sqrt(2) than sqrt2.