Prove Q[sqrt 2, sqrt 3] is a field.

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In summary, to prove that Q[sqrt 2, sqrt 3] is a field, we need to show that each element in this set has an inverse. To find the inverse, we can use the formula (a - b*sqrt n)/(a^2 - n*b^2) for elements in Q[sqrt n], where a and b are elements of Q[sqrt n]. Since an element in Q[sqrt 2, sqrt 3] can be written in the form a + b*sqrt 2 + c*sqrt 3 + d*sqrt 6, where a, b, c, and d are elements in Q[sqrt 2], we can use the same formula for finding the inverse
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gotmilk04
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Homework Statement


Prove Q[sqrt 2, sqrt 3] is a field.


Homework Equations


Q[sqrt 2, sqrt 3]= {r + s[tex]\sqrt{2}[/tex] + t[tex]\sqrt{3}[/tex] + u[tex]\sqrt{6}[/tex]| r,s,t,u[tex]\in Q[/tex]}

The Attempt at a Solution


I know I have to show each element has an inverse, but I don't know how on these elements.
 
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To find the inverse of something in Q[[tex]\sqrt{2}, \sqrt{3}[/tex]] it might be helpfull to first notice the inverse of something in Q[[tex]\sqrt{2}[/tex]].

(a + b[tex]\sqrt{2}[/tex])^-1 = (a - b[tex]\sqrt{2}[/tex])/(a^2 - 2*b^2)

In general (a + b*sqrt n)^-1 = (a - b*sqrt n)/(a^2 - n*b^2)

Next notice that an element of Q[[tex]\sqrt{2}[/tex],[tex]\sqrt{3}[/tex]] is also of the form a + b[tex]\sqrt{3}[/tex] where a and b are elements of Q[[tex]\sqrt{2}[/tex]] (i.e. they are each a + b*[tex]\sqrt{2}[/tex]) so an element of Q[[tex]\sqrt{2}[/tex], [tex]\sqrt{3}[/tex]] is like ((a + b[tex]\sqrt{2}[/tex]) + (c + d[tex]\sqrt{2}[/tex])[tex]\sqrt{3}[/tex]) = a + b[tex]\sqrt{2}[/tex] + c[tex]\sqrt{3}[/tex] + d[tex]\sqrt{6}[/tex].

So the inverse of such an element is still (a - b[tex]\sqrt{3}[/tex])/(a^2 - 3*b^2) where the a and b in this case are elements of Q[[tex]\sqrt{2}[/tex]] (and so of the form a + b[tex]\sqrt{2}[/tex]).
 
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1. What is a field in mathematics?

A field is a mathematical structure consisting of a set of elements, along with operations of addition, subtraction, multiplication, and division, that follow certain axioms or rules.

2. What does it mean for Q[sqrt 2, sqrt 3] to be a field?

It means that the set Q[sqrt 2, sqrt 3], which consists of all numbers that can be expressed as a combination of rational numbers and the square roots of 2 and 3, follows all the rules of a field.

3. What are the axioms or rules that Q[sqrt 2, sqrt 3] must follow to be considered a field?

The axioms for a field include closure under addition and multiplication, commutativity and associativity of addition and multiplication, existence of additive and multiplicative identities, existence of additive and multiplicative inverses, and distributivity of multiplication over addition.

4. How can we prove that Q[sqrt 2, sqrt 3] is a field?

We can prove that Q[sqrt 2, sqrt 3] is a field by showing that it satisfies all the axioms or rules of a field. This can be done by verifying that all the elements in Q[sqrt 2, sqrt 3] have an additive and multiplicative inverse, that the operations of addition and multiplication follow the necessary properties, and that the set is closed under these operations.

5. What are some real-world applications of fields in mathematics?

Fields have many real-world applications, including in engineering, physics, and computer science. They are used to model and solve various problems involving quantities such as force, voltage, and data transmission. They are also essential in cryptography and coding theory, which have numerous applications in secure communication and data storage.

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