Set of algebraic integer form a ring.

Question

An algebraic integer is a complex number that is the root of monic polynomial with integer coefficients. Show that the set of algebraic integers is a subring of $C$. (Hint: Use symmetric function theorem).

I also know that $\alpha \in$ $C$ is an algebraic integer if and only if $m_\alpha,_Q \in Z[x].$

Thanks for the help.

Answer 1

Let $\alpha,\beta\in\mathbb{C}$ be algebraic integers; e.g. $\alpha^3+\alpha+1=0$ and $\beta^2+1=0$.

Since the polynomials that $\alpha$ and $\beta$ satisfy are monic, we can, by division, write any element of $\mathbb{Z}[\alpha,\beta]$ as an integer combination of elements of the basis, $B$: $$ \left\{1,\alpha,\alpha^2,\beta,\alpha\beta,\alpha^2\beta\right\}\tag1 $$ That is, we can write any $p\in\mathbb{Z}[\alpha,\beta]$ as $$ p=\overbrace{\begin{bmatrix}p_0&p_1&p_2&p_3&p_4&p_5\end{bmatrix}}^{\mathbb{Z}^n}\overbrace{\ \begin{bmatrix}1\\\alpha\\\alpha^2\\\beta\\\alpha\beta\\\alpha^2\beta\end{bmatrix}\ }^B\tag2 $$ which means we can write $$ \begin{align} p\ \begin{bmatrix}1\\\alpha\\\alpha^2\\\beta\\\alpha\beta\\\alpha^2\beta\end{bmatrix} &=\overbrace{\begin{bmatrix} p_{0,0}&p_{0,1}&p_{0,2}&p_{0,3}&p_{0,4}&p_{0,5}\\ p_{1,0}&p_{1,1}&p_{1,2}&p_{1,3}&p_{1,4}&p_{1,5}\\ p_{2,0}&p_{2,1}&p_{2,2}&p_{2,3}&p_{2,4}&p_{2,5}\\ p_{3,0}&p_{3,1}&p_{3,2}&p_{3,3}&p_{3,4}&p_{3,5}\\ p_{4,0}&p_{4,1}&p_{4,2}&p_{4,3}&p_{4,4}&p_{4,5}\\ p_{5,0}&p_{5,1}&p_{5,2}&p_{5,3}&p_{5,4}&p_{5,5} \end{bmatrix}}^{\mathbb{Z}^{n\times n}} \begin{bmatrix}1\\\alpha\\\alpha^2\\\beta\\\alpha\beta\\\alpha^2\beta\end{bmatrix}\tag3\\[6pt] pB&=MB\tag4 \end{align} $$ Since $M$ satisfies its characteristic polynomial $$ {\large\chi}_M(M)=0\tag5 $$ we have that $p$ also satisfies ${\large\chi}_M$: $$ {\large\chi}_M(p)=0\tag6 $$ Since ${\large\chi}_M$ is a monic integer polynomial, $p$ must be an algebraic integer.

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Example 1

For example, $$ (\alpha+\beta)\begin{bmatrix}1\\\alpha\\\alpha^2\\\beta\\\alpha\beta\\\alpha^2\beta\end{bmatrix}=\begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ -1 & -1 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & -1 & -1 & 0 \end{bmatrix}\begin{bmatrix}1\\\alpha\\\alpha^2\\\beta\\\alpha\beta\\\alpha^2\beta\end{bmatrix} $$ and ${\large\chi}_M(\lambda)=\det(M-I\lambda)=\lambda^6+5\lambda^4+2\lambda^3+4\lambda^2-4\lambda+1$.

Thus, $\alpha+\beta$ is an algebraic integer.

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Example 2

Furthermore, $$ \alpha\beta\ \begin{bmatrix}1\\\alpha\\\alpha^2\\\beta\\\alpha\beta\\\alpha^2\beta\end{bmatrix}=\begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & -1 & -1 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix}1\\\alpha\\\alpha^2\\\beta\\\alpha\beta\\\alpha^2\beta\end{bmatrix} $$ and ${\large\chi}_M(\lambda)=\det(M-I\lambda)=\lambda^6-2\lambda^4+\lambda^2+1$.

Thus, $\alpha\beta$ is an algebraic integer.

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Conclusions

The key points that can be extrapolated from the examples above are that

(1) since $\alpha$ and $\beta$ both satisfy monic integer polynomials, $\mathbb{Z}[\alpha,\beta]$ can be written as a $\mathbb{Z}$-module generated by the Kronecker product (or flattened outer product) of $\left\{1,\alpha,\alpha^2,\dots,\alpha^{k-1}\right\}$ and $\left\{1,\beta,\beta^2,\dots,\beta^{m-1}\right\}$.

(2) the action of $p\in\mathbb{Z}[\alpha,\beta]$ on the basis of $n=km$ generators, from the Kronecker product above, can be represented by $M\in\mathbb{Z}^{n\times n}$.

(3) since $M$ satisfies its own characteristic polynomial, ${\large\chi}_M(M)=0$, so does $p$, ${\large\chi}_M(p)=0$

(4) the characteristic polynomial of an integer matrix is a monic integer polynomial.

Therefore, if $\alpha$ and $\beta$ are algebraic integers, then each $p\in\mathbb{Z}[\alpha,\beta]$ is an algebraic integer.