(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the collection F of all real numbers of the form x+y√2, where x and

y are rational numbers. Prove (by direct verification) that F satisfies all the field

axioms (just like R) under the usual addition and multiplication.

2. Relevant equations

Field axioms: There exist two binary operations, called addition + and

multiplication ∗, such that the following hold:

1) commutativity x + y = y + x, xy = yx

2) associativity x + (y + z) = (x + y) + z, x(yz) = (xy)z

3) distributivity x(y + z) = xy + xz

4) Existence of 0; 1 such that x + 0 = x, 1 · x = x,

5) Existence of negatives: For every x there exists y such that x + y = 0.

6) Existence of reciprocals: For every x ̸= 0 there exists y such that

xy = 1.

3. The attempt at a solution

I just want to make sure I did this right. If you were to prove that a collection F ( a collection of all real #'s of the form x+y[tex]\sqrt{2}[/tex] where x & y are rational numbers ) satisfies all of the field axioms by direct verification, would you just do something like suppose m,n,o belong to F. then m=x_{1}+y_{1}[tex]\sqrt{2}[/tex], etc. and then you just say m+(n+o) = x_{1}+y_{1}[tex]\sqrt{2}[/tex] + ..... until you return to m+n+o = (m+n)+o ??? And then proceed to do so for all the axioms mentioned above? It's supposed to be really trivial right?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Prove that F satisfies all field axioms by method of direct verification

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