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This is always true for every positive integer $n$. If ${\beta }^n$ were algebraic, then $\beta$ would be a root of the polynomial $X^n-{\beta }^n$ over the field of algebraic numbers. By transitivity of algebraic extensions, $\beta$ would then be algebraic over $\mathbb{Q}$, contradicting its transcendence.

LetIf $x$ were rational, thenwould make $\sqrt[3]{6}=(x^3-5)/(3x)$ rational. But if $\sqrt[3]{6}=a/b$ in lowest terms, then $a^3=6b^3$, which is impossible by unique prime factorization: the exponent of $2$ on the two sides is respectively a multiple of three and one more than a multiple of three. Therefore

Use the given convergent seriesSuppose this number were $a/b\in \mathbb{Q}$. Choose $N$ large enough that $b\mid (2N+1)!$. Multiplication by $(2N+1)!$ makes both the rational number and the partial sum through $n=N$ integers. Their differencewould therefore be an integer. It is positive, whilefor sufficiently large $N$, a contradiction. Hence

A transcendental number is a complex number that is not a root of any nonzero polynomial with rational, equivalently integer, coefficients. LetIf $a\ne 0$ and $y$ were an algebraic number, then $e$ would satisfyover the algebraic extension $\mathbb{Q}(y)$. By transitivity of algebraic extensions, $e$ would be algebraic over $\mathbb{Q}$, contradicting its transcendence. If $a=0$, then $b\ne 0$ and $e=b/y$ would again be algebraic. Thus

An algebraic closure of a field $K$ is an algebraically closed field that is an algebraic extension of $K$.

By definition, every element ofis algebraic over $\mathbb{Q}$, and the question allows us to use that this set is a field. It remains to prove that it is algebraically closed.

Let $f\in \overline{\mathbb{Q}}[X]$ be nonconstant. Its finitely many coefficients generate a finite extension $E/\mathbb{Q}$. By the fundamental theorem of algebra, $f$ has a root $z\in \mathbb{C}$. Since $z$ is a root of a polynomial over $E$, it is algebraic over $E$; since $E/\mathbb{Q}$ is algebraic, transitivity of algebraic extensions makes $z$ algebraic over $\mathbb{Q}$. Thus $z\in \overline{\mathbb{Q}}$. Repeating after division by $X-z$ shows that $f$ splits there, so $\overline{\mathbb{Q}}$ is algebraically closed and is an algebraic closure of $\mathbb{Q}$.