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Examples of algebraic numbers invariably involve expressions containing integers, or even rational numbers, together with radicals—that is, nth roots, possibly for various values of n. Examples include 2\sqrt{2}, the golden ratio
1+52, 2+3, 3−23+1523−1…\frac{1+\sqrt{5}}{2},\, \sqrt{2}+\sqrt{3},\, \frac{\sqrt[5]{\sqrt{3-\sqrt[3]{2}+1}}}{\sqrt[3]{2}-1}\ldots
As a result, many people think that every algebraic number can be expressed in radicals. In fact, nothing could be further from the truth!
For example, the equation x5 − x − 1 = 0 has five roots: one is real, while the other four are complex and form two conjugate pairs. Yet none of these roots can be expressed in radicals! Nevertheless, all five are algebraic numbers—and even algebraic integers.