lostNfound said:I'm going through a abstract algebra book I found and am trying to learn more about group theory by going through some of the proofs and practice sets, but am having trouble with the following:
Prove that G={a+b*sqrt(2) | a,b E R; a,b not both 0} is a group under ordinary multiplication.
Any help would be awesome!
lostNfound said:So I know that in order to prove it is a group, there are several things that have to be confirmed, including associativity, closure, inverses, and identity. Associativity seems like it can be assumed, but the others till have to be proved. I'm having most of my trouble with the closure and inverse proofs, especially that for closure.
Why did you delete this? I wouldn't say you should assume associativity but certainly you can just note that multiplication of real numbers is associative and this is just a subset of the real numbers. To show closure write the product of two such numbers as [itex](a+ b\sqrt{2})(c+ d\sqrt{2})[/itex] and actually do the multiplication. What do you get? Show that it can be written as [itex]u+ v\sqrt{2}[/itex] by showing what u and v must be. The identity is [itex]1= 1+ 0\sqrt{2}[/itex], of course.So I know that in order to prove it is a group, there are several things that have to be confirmed, including associativity, closure, inverses, and identity. Associativity seems like it can be assumed, but the others till have to be proved. I'm having most of my trouble with the closure and inverse proofs, especially that for closure.
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