I'm working independently through the textbook Algebra by Cohn and have reached the section on elementary symmetric functions. I thought I had understood them but am confused by this question:
Express $\Sigma x_1^3 x_2 x_3$ in terms of elementary symmetric functions.
My answer: $e_3(e_1^2-2e_2)$
Book's answer: $e_3(e_1^2-2e_2)-e_4e_1$
Where does $e_4$ come from? As there are only three indeterminates in the question, I don't understand why I need to use $e_4$ at all?
My approach to solving the question was to write out $\Sigma x_1^3 x_2 x_3$ as $x_1^3x_2 x_3+x_1 x_2^3 x_3+x_1 x_2 x_3^3 = x_1x_2x_3(x_1^2+x_2^2+x_3^2)$ and then notice that the term in brackets is $e_1^2-2e_2$ and that $x_1x_2x_3 = e_3$