More intro abstract algebra problems

In summary, the author proves that 1 is the multiplicative identity and also that there exists an inverse of A such that A×A-1 = 1. However, in order to solve for A-1, they need to know that √2 is irrational.
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Homework Statement



Define the set Q[√2] to be the set {a + b√2 | a, b are rationals}, and define addition and multiplication as "usual" (so 2×4 = 8, 2 + 4 = 6, you know, the usual). Show that for any nonzero A in the set Q[√2], there exists an inverse element so that A×A-1 = 1Q[√2].

There is a hint saying that I'll need the fact that √2 is irrational. He asks "where" will I need this fact?

Homework Equations



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The Attempt at a Solution



So I proved that the number 1 itself is the multiplicative identity, and it is also definitely in this set Q[√2]. Now I need to show that there exists an inverse of A so that A×A-1 = 1. And of course A-1 must also be an element of Q[√2].

so let q = a + b√2, then find q-1 = c + d√2 so that

(a + b√2)(c + d√2) = 1.

So we have
ac + ad√2 + cb√2 + 2bd = 1
ac + 2bd + (ad + cb)√2 = 1.

Since √2 is irrational, α + β√2 = 1 implies α = 1, β = 0. This is where i need the fact that √2 is irrational (although i needed it to show 1 = 1Q[√2])

So ad + cb = 0 and ac + 2bd = 1. But how do show that solutions must always exist for these two equations, if I only have two equations but four unknowns? Am i even on the right track here?
 
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  • #2
This appears to be a fairly advanced class so surely you are already familiar with "rationalizing the denominator"! If [itex]x= 1/(a+b\sqrt{2})[/itex] then
[tex]x= \dfrac{1}{a+ b\sqrt{2}}\dfrac{a-b\sqrt{2}}{(a- b\sqrt{2}}= \dfrac{a- b\sqrt{2}}{a^2- 2b^2}[/tex]
Of course, both [itex]a/(a^2- 2b^2)[/itex] and [itex]-b/(a^2- 2b^2)[/itex] are rational as long as [itex]a^2- 2b^2[/itex] is not 0! That is the part that requires that [itex]\sqrt{2}[/itex] not be rational.
 
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  • #3
Of course, the first thing i thought it was the reciprocal, but sometimes I get so worked up in these so-called "advanced" courses that i forget such simple things i learned in high school. I guess it's time i put it all together.

Thanks very much for your help, HallsofIvy!

Edit: while this may be an advanced class, the stuff we are doing is far from advanced!
 
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1. What is abstract algebra?

Abstract algebra is a branch of mathematics that deals with the study of algebraic structures, such as groups, rings, and fields. It focuses on the properties and relationships between mathematical objects rather than specific numbers or calculations.

2. Why is abstract algebra important?

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3. What are some common topics in abstract algebra?

Some common topics in abstract algebra include group theory, ring theory, field theory, and linear algebra. Other topics may include Galois theory, representation theory, and homological algebra.

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