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Magazine Problem 1
In Exercises 1 through 5 , show that the given number \(\alpha \in \mathrm{C}\) is algebraic over \(Q\) by finding \(f(x) \in \mathbb{Q}[x]\) such that \(f(\alpha)=0 .\) $$ 1+\sqrt{2} $$
Problem 2
In Exercises 1 through 5 , show that the given number \(\alpha \in \mathrm{C}\) is algebraic over \(Q\) by finding \(f(x) \in \mathbb{Q}[x]\) such that \(f(\alpha)=0 .\) $$ \sqrt{2}+\sqrt{3} $$
Problem 2
\(\sqrt{2}+\sqrt{3}\)
Problem 3
In Exercises 1 through 5 , show that the given number \(\alpha \in \mathrm{C}\) is algebraic over \(Q\) by finding \(f(x) \in \mathbb{Q}[x]\) such that \(f(\alpha)=0 .\) $$ 1+i $$
Problem 4
In Exercises 1 through 5 , show that the given number \(\alpha \in \mathrm{C}\) is algebraic over \(Q\) by finding \(f(x) \in \mathbb{Q}[x]\) such that \(f(\alpha)=0 .\) $$ \sqrt{1+\sqrt[3]{2}} $$
Problem 5
In Exercises 1 through 5 , show that the given number \(\alpha \in \mathrm{C}\) is algebraic over \(Q\) by finding \(f(x) \in \mathbb{Q}[x]\) such that \(f(\alpha)=0 .\) $$ \sqrt{\sqrt{2}-i} $$
Problem 6
In Exercises 6 through 8 , find \(\operatorname{irr}(\alpha, Q)\) and \(\operatorname{deg}(\alpha, Q)\) for the given algebraic number \(\alpha \in \mathbb{C}\). Be prepared to prove that your polynomials are irreducible over \(\mathrm{Q}\) if challenged to do so. $$ \sqrt{3-\sqrt{6}} $$
Problem 6
\(\sqrt{3-\sqrt{6}}\)
Problem 7
\(\sqrt{\left(\frac{1}{3}\right)+\sqrt{7}}\)
Problem 8
In Exercises 6 through 8 , find \(\operatorname{irr}(\alpha, Q)\) and \(\operatorname{deg}(\alpha, Q)\) for the given algebraic number \(\alpha \in \mathbb{C}\). Be prepared to prove that your polynomials are irreducible over \(\mathrm{Q}\) if challenged to do so. $$ \sqrt{2}+i $$
