is an ordered field with the order inherited from because we verify:
Fernando Revilla
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!
is an ordered field with the order inherited from because we verify:
Fernando Revilla
So, . (One usually says "s is in Q," not "s exists in Q.") Suppose . Then . Checking point 1 is easier.
is a subfield of : every element of K is a real number and satisfies the axioms of a field.(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
With the order inherited from the set is an ordered field: with the regular order, 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 and : .
What is "it" in your question? The notation is used to denote a set that has all possible elements of the form for and only such element. So, something of the form is in this set, called K here, and every element of K has this form.
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 . Yes, if , then y has the form for some rational .Also, would i make y (in the original problem it is xsub1) ssub1 + tsub1sqrt2 ?
Is is not clear to me what s and t are, how they were introduced. In any case, if are rational, then yes, and , by definition of K. I am not sure how this advanced the proof, though.and then say that s+tsqrt2,ssub1 + tsub1sqrt2 is in k?
To answer my question from the previous post: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 , there exists such that and . Note that I introduced by saying "there exist". Also, the point was not to say that , which is a triviality; the point was to say that . After that, you can see what x + y looks like.What should the first phrase of the proof be?
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.