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In Art. 535 of Higher Algebra by Hall and Knight it is written,

Let $p_0 x^n+p_1 x^{n-1}+p_2 x^{n-2}+\dotsm+p_{n-1} x^{1}+p_n$ be a rational integral function of $x$ of $n$ dimensions and let us denote it by $f(x)$; then $f(x)=0$ is the general type of a rational integral equation of the $n^{th}$ degree. Dividing throughout by $p_0$, we see that without any loss of generality we may take $x^n+p_1 x^{n-1}+p_2 x^{n-2}+\dotsm+p_{n-1} x^{1}+p_n$ as the type of a rational integral equation of any degree.

Questions:

(i)What is the meaning of the phrase "without any loss of generality"?

(ii)What is the meaning of the phrase "rational integral function"?

(iii)If the equation $p_0 x^n+p_1 x^{n-1}+p_2 x^{n-2}+\dotsm+p_{n-1} x^{n- 1}+p_n$ is divided by $p_0$ throughout, we obtain $x^n+\frac{p_1}{p_0} x^{n-1}+\frac{p_2}{p_0} x^{n-2}+\dotsm+\frac{p_{n-1}}{p_0} x^{n-1}+\frac{p_n}{p_0}$ but in the Art. it is written that on "Dividing throughout by $p_0$, we see that without any loss of generality we may take $x^n+p_1 x^{n-1}+p_2 x^{n-2}+\dotsm+p_{n-1} x^{n-1}+p_n$ as the type of a rational integral equation of any degree." Should not the coefficients of $x$ starting from $x^{n-1}$ be $\frac{p_1}{p_0},\frac{p_2}{p_0},\dotsm,\frac{p_n}{p_0}$ instead of $p_1,p_2,\dotsm p_n$.

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I can definitely help you with the first two questions, but I believe the book is misleading with their explanation.

(i) Without Loss of Generality is a phrase which denotes that an assumption on a problem has been made, but this will not introduce any new restrictions. Usually, it is used in proofs. Short-hand is w.l.o.g. Note that the word 'any' in your book is very unnecessary.

Here's an example from Wikipedia that I like:

Prove that if there are three objects painted either red or blue, then there must be at least two objects of the same color.

Then, w.l.o.g, assume the first object is blue. If either of the other two objects is blue, we are finished; if not, the other two objects must both be red and we are still finished. Without loss of generality is used here because we could have assumed the first object is red, but this would not have affected the way we finished the proof. Other times you can use w.l.o.g is with symmetrical inequalities or variables.

(ii) A rational integral function is just a fancy way of saying polynomial with rational coefficients. It may also denote a polynomial with integer coefficients. To be honest, it is rarely used today, and it is much easier and clearer to say a polynomial with rational or integer coefficients.

(iii) I think the book assumes a new value of $p_1$, $p_2$, $p_3...$ when dividing $f(x)$ by $p_0$. Don't take my word for it, but that is how I interpret it.

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