Question

If $\mathbf{a}\mathbf{\in }\mathbf{\mathbb{Q}}$is an algebraic integer, as defined on page 350, show that$\mathbf{a}\mathbf{\in }\mathbf{\mathbb{Z}}$.

Short answer

It has been proved that $a\in \mathrm{\mathbb{Z}}$.

Step-by-step solution

Define algebraic integer

Given that$\mathrm{a}\in \mathrm{\mathbb{Q}}$is an algebraic integer.

An algebraic integer is a complex number that is the root of some monic polynomial with integer coefficients.

Prove that  a∈ℤ

Since $\mathrm{a}\in \mathrm{\mathbb{Q}}$, let $a=r/s$where $r,s\in \mathbb{Z}$and are co primes.

Since a is an algebraic integer thereforea is a root of some monic polynomial

$x^n+c_{n-1}{x}^{n-1}+\mathrm{...}+c_0$where all $c_j\in \mathbb{Z}$.

By Rational Root Test, $s|1$. So,$s=\pm 1$.

Thus,a is an integer.

Conclusion

Thus, it can be concluded that $\mathrm{a}\in \mathrm{\mathbb{Z}}$.