# Show √ 2 + √ 3 algebraic over Q

• HaLAA
In summary, a number is algebraic over Q if it can be expressed as a finite combination of rational numbers using the operations of addition, subtraction, multiplication, and division. √2 + √3 is algebraic over Q as it is a root of the polynomial equation x^4 - 10x^2 + 1 = 0. Its degree over Q is 4. Irrational numbers can also be algebraic over Q as long as they are roots of polynomial equations with rational coefficients. However, not all numbers are algebraic over Q and some, like π and e, are known as transcendental numbers.
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## Homework Statement

Show √ 2 + √ 3 algebraic over Q. Find its degree over Q. Prove the answer.

## The Attempt at a Solution

Let ##\alpha= \sqrt{2}+\sqrt{3}\in \mathbb{R}##, then ##\alpha^4-10\alpha^2+1=0## which is a root of ##f(x)=x^4-10x^2+1## where ##f(x)## in ##\mathbb{Q}[X]##. Apply rational root test, ##f(\pm 1)=-8## which implies ##\alpha \notin \mathbb{Q}##.
Also, ##f(x)=x^4-10x^2+1=(x^2+2\sqrt{6}-5)(x^2-2\sqrt{6}+5)\notin \mathbb{Q}[X]##, hence ##f(x)## is irreducible in ##\mathbb{Q}[X]## which shows ##\alpha## is degree of ##4##.

Looks right.

## What does it mean for a number to be algebraic over Q?

A number is algebraic over Q if it is a root of a polynomial equation with rational coefficients. In other words, it can be expressed as a finite combination of rational numbers using the operations of addition, subtraction, multiplication, and division.

## Is √2 + √3 algebraic over Q?

Yes, √2 + √3 is algebraic over Q. This can be proven by finding a polynomial equation with rational coefficients that has √2 + √3 as a root. In this case, the equation x^4 - 10x^2 + 1 = 0 has √2 + √3 as one of its roots.

## What is the degree of √2 + √3 over Q?

The degree of √2 + √3 over Q is 4. This can be determined by looking at the polynomial equation that has √2 + √3 as a root. In this case, the degree is 4 because the highest power of x in the equation is 4.

## Can irrational numbers be algebraic over Q?

Yes, irrational numbers can be algebraic over Q. As long as they are roots of polynomial equations with rational coefficients, they are considered algebraic over Q.

## Are all numbers algebraic over Q?

No, not all numbers are algebraic over Q. For example, numbers such as π and e are not algebraic over Q, as they cannot be expressed as a finite combination of rational numbers using the operations of addition, subtraction, multiplication, and division. These numbers are known as transcendental numbers.

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