Proving an Algebraic Number: Root 3 + Root 2

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In summary, an algebraic number is a number that can be expressed as a root of a polynomial equation with integer coefficients. Root 3 + Root 2 is an algebraic number that can be expressed as the sum of the square roots of 3 and 2. To prove that it is an algebraic number, we need to construct a polynomial with Root 3 + Root 2 as a root, verify that the coefficients are integers, and show that the polynomial has no rational roots. The Rational Root Theorem can then be used to prove that the polynomial is irreducible, leading to the conclusion that Root 3 + Root 2 is an algebraic number. Proving this is important as it helps us understand the properties of
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lokisapocalypse
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I need to show that this is an algebraic number.

In other words,

I need to show: an*x^n + an1*x^(n-1) + ... + a1 * x^1 + a0 * x^0 =
where the a terms are not ALL 0 but some of them can be.

Like for root 2 by itself,

I have 1 * (root 2) ^ 2 + 0 * (root 2)^1 + -2 * (root 2) ^ 0
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Wtf I Just Replied To This In The Homework Section
 
  • #3
But did you reply to it the in Linear algebra section too?
 

Related to Proving an Algebraic Number: Root 3 + Root 2

What is an algebraic number?

An algebraic number is a number that can be expressed as a root of a polynomial equation with integer coefficients.

What is Root 3 + Root 2?

Root 3 + Root 2 is an algebraic number that can be expressed as the sum of the square roots of 3 and 2.

How do you prove that Root 3 + Root 2 is an algebraic number?

To prove that Root 3 + Root 2 is an algebraic number, we need to show that it satisfies a polynomial equation with integer coefficients. This can be done by constructing a polynomial with Root 3 + Root 2 as a root, and then verifying that the coefficients are integers.

What are the steps to prove that Root 3 + Root 2 is an algebraic number?

The steps to prove that Root 3 + Root 2 is an algebraic number are as follows:

  1. Construct a polynomial with Root 3 + Root 2 as a root.
  2. Verify that the coefficients of the polynomial are integers.
  3. Show that the polynomial has no rational roots.
  4. Use the Rational Root Theorem to prove that the polynomial is irreducible.
  5. Conclude that Root 3 + Root 2 is an algebraic number.

Why is proving Root 3 + Root 2 an algebraic number important?

Proving that Root 3 + Root 2 is an algebraic number is important because it helps us understand the properties and behavior of algebraic numbers. It also allows us to solve various mathematical problems and equations involving Root 3 + Root 2. Furthermore, it is a fundamental concept in algebra and number theory.

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