Question

Extend Exercise 20 to any finite number of elements.

Short answer

Any finite number of elements have a greatest common divisor that can be written as a linear combination of the elements.

Step-by-step solution

common divisor of finite elements

Let $R$ be a PID and $a_1,a_2,\dots ,a_n\in R$.Consider the principal ideals $\left(a_1\right),\left(a_2\right),\dots \left(a_n\right)\in \mathrm{ℝ}$ then $\left(a_1\right)+\left(a_2\right)+\dots +\left(a_n\right)$ is an ideal.

Since, $R$ is a PID, there exists an element in $x$ such that $\left(a_1\right)+\left(a_2\right)+\dots +\left(a_n\right)=x$.

Thus,$\left(a_1\right)+\left(a_2\right)+\dots +\left(a_n\right)=x$ implies $\left(a_1\right)\subset x,\left(a_2\right)\subset x,\dots ,\left(a_n\right)\subset x$ and again, $\left(a_i\right)\subset x$ implies$x|a_i$ for all $i=\{1,2,\dots ,n\}$.

Therefore, x is a common divisor of $a_1,a_2,\dots ,a_n\in R$.

gcd of a and b

Let c be a another common divisor of $a_1,a_2,\dots ,a_n\in R$then $c|a_i$ implies that $\left(a_i\right)\subset \left(c\right)$ for all $i=\{1,2,\dots ,n\}$.

Since, $\left(a_1\right)+\left(a_2\right)+\dots +\left(a_n\right)$is the smallest ideal then $\left(a_1\right)+\left(a_2\right)+\dots +\left(a_n\right)\subset \left(c\right)$ which can be written as $\left(x\right)\subset \left(c\right)$. Thus, $c|x$ and x is the gcd of $a_1,a_2,\dots ,a_n\in R$.

Since, $\left(x\right)=\left(a_1\right)+\left(a_2\right)+\dots +\left(a_n\right)$ then the linear combination can be written as $x=\sum _{i=1}^n{a}_i{m}_i$ where $m_i\in \mathrm{ℝ}$.

Therefore,any finite number of elements have a greatest common divisor that can be written as a linear combination of the elements.