Question

How do you prove that the set of roots of polynomial equations in one variable with integer coefficients is algebraically closed?

Answer 1

Hint: We need to use the basic algebraic concepts to prove this given question. We assume a general form for all the roots of the polynomial equation in one variable and then take a product of all the possible variations of the polynomial equation after substituting the roots. These coefficients should be rational to say that this set of roots of the polynomial equations in one variable are algebraically closed.

Complete step by step solution:
Let us consider the polynomial equation in one variable with integer coefficients as given by $P\left( x \right).$
$\Rightarrow P\left( x \right)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n-2}}{{x}^{n-2}}+\ldots +{{a}_{0}}$
In the above equation, x is the variable in the polynomial equation, n is the degree of the polynomial equation, ${{a}_{0}},{{a}_{1}},\ldots ,{{a}_{n-1}},{{a}_{n}}$ are the integer coefficients. An important point to note is that in the above equation, ${{a}_{n}}=1.$ Therefore, the equation reduces to,
$\Rightarrow P\left( x \right)={{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n-2}}{{x}^{n-2}}+\ldots +{{a}_{0}}$
Let these coefficients be denoted generally in the form ${{a}_{i}}.$ For each value of i from 0 to n-1, let ${{P}_{i}}\left( x \right)$ be a polynomial equation with integer coefficients such that ${{P}_{i}}\left( {{a}_{i}} \right)$ is zero, that is the given root satisfies the polynomial equation. We denote the roots of ${{P}_{i}}\left( x \right)=0$ by ${{a}_{i,1}},{{a}_{i,2}},\ldots ,{{a}_{i,{{n}_{i}}}}.$
 Let us consider the product of all such variations of $P\left( x \right)$ and we get the following equation,
$\Rightarrow R\left( x \right)=\prod\limits_{{{j}_{n-1}}=1}^{{{n}_{n-1}}}{\prod\limits_{{{j}_{n-2}}=1}^{{{n}_{n-2}}}{\ldots \prod\limits_{{{j}_{0}}=1}^{{{n}_{0}}}{\left( {{x}^{n}}+{{a}_{n-1,{{j}_{n-1}}}}{{x}^{n-1}}+\ldots +{{a}_{0,{{j}_{0}}}} \right)}}}$
Here, if we expand this given the coefficients of $P\left( x \right)$ , we can see that the coefficients of $R\left( x \right)$ is symmetric for all the roots for each of the polynomial equation ${{P}_{i}}\left( x \right).$ Knowing this, we can confirm that the coefficients of $R\left( x \right)$ are rational and they include the roots of $P\left( x \right).$
Therefore, the roots of the polynomial expression are algebraically closed.

Note:It is essential to know the meaning of the terms algebraically closed and polynomial equations in one variable with integer coefficients. Algebraically closed means that for every non constant polynomial in a function, it has a root in the same function. Here, we have shown the same, that is $R\left( x \right)$ having all the non-constant polynomials for the function $P\left( x \right)$ , has roots in $P\left( x \right).$ Polynomial equations in one variable with integer coefficients means that it is an equation with a variable degree n and all the coefficients being an integer in one variable, say x.