I would assume that \(\sqrt[31]{12}+\sqrt[12]{31}\) is rational and try to find a contradiction

However, I don't know where to start. Can someone give me a tip on how to approach this problem?

Jillian Edgerton

Answered 2022-01-08
Author has **1651** answers

Let \(\displaystyle{\mathbb{{Q}}}{\left(\alpha\right)}\) denote the smallest field containing \(\displaystyle{\mathbb{{Q}}}\) and \(\displaystyle\alpha\)

The theory of field extensions tells us that \(\mathbb Q(\sqrt[31]{12})\) has degree \(\displaystyle{31}\) over \(\displaystyle{\mathbb{{Q}}}\), \(\mathbb Q(\sqrt[12]{31})\) has degree 12 over \(\displaystyle{\mathbb{{Q}}}\) , and, because \(\displaystyle{\left({31},{12}\right)}={1}\), we have

\(\mathbb Q(\sqrt[31]{12})\cap\mathbb Q(\sqrt[12]{31})=\mathbb Q\)

If \(\sqrt[31]{12}+\sqrt[12]{31}\) were a rational number, we would have

\(\sqrt[31]{12} \in \mathbb Q(\sqrt[31]{12})\cap\mathbb Q(\sqrt[12]{31})=\mathbb Q\) But \(\sqrt[31]{12}\) is not rational, contradiction.

Vasquez

Answered 2022-01-11
Author has **8850** answers

It is known that algebraic integers are closed under addition, subtraction, product and taking roots.

Since 12 and 31 are algebraic integers, so does their roots \(\sqrt[31]{12},\sqrt[12]{31}\). Being the sum of two such roots, \(\sqrt[31]{12}+\sqrt[12]{31}\) is an algebraic integer.

It is also known that if an algebraic integer is a rational number, it will be an ordinary integer. Notice

\(2<\sqrt[31]{12}+\sqrt[12]{31}<\sqrt[31]{2^4}+\sqrt[12]{2^5}=2^{\frac{4}{31}}+2^\frac{5}{12}<2\sqrt2<3\)

\(\sqrt[31]{12}+\sqrt[12]{31}\) isn't an integer and hence is an irrational number.

asked 2021-09-15

Prove or disprove that the product of two irrational numbers is irrational.

asked 2022-01-06

Can an irrational number raised to an irrational power be rational?

If it can be rational, how can one prove it?

If it can be rational, how can one prove it?

asked 2022-01-04

How can I prove that the sum \(\sqrt{2}+\sqrt[3]{3}\) is an irrational number ??

asked 2021-12-14

What is the square root of 80?

asked 2021-12-16

What is the square root of 216?

asked 2021-12-17

What is the square root of 24?

asked 2021-12-06

Prove that 1/3 is rational.