Polynomial of minimum degree definition?

Question

Theorem:

Let $ F$ be a field, $I$ a nonzero ideal in $F [x]$, and $g(x)$ an element of $F[x]$. Then, $I = \langle g(x) \rangle$ if and only if $g(x)$ is a nonzero polynomial of minimum degree in $I$.

I'm reading Gallian's abstract algebra and have come across the term polynomial of minimum degree but have not seen it before this point. What is a polynomial of minimum degree?

Answer 1

It is a nonzero polynomial $g \in I$ such that the degree of $g$ is the smallest possible among the degrees of nonzero polynomials in $I$.

Or more formally, $g$ is a nonzero polynomial of minimum degree in $I$ when $g \in I, g \neq 0$, and if $h \in I$ with $h \neq 0$ then $\deg(h) \ge \deg(f)$

Answer 2

If one wanted to say "the richest person in the Britain except the Queen" (I suppose this is easy enough to understand) in a mathematical fashion, one might say "a non-monarch inhabitant of the United Kingdom of maximum income". The phrase you are asking about is quite similar to that, so you can work out for yourself what it means. Note the the latter description allows for more than one person being designated, in case of a tie in wealth; similarly there may be (in fact will be) multiple polynomials of minimum degree to choose from (although there is exactly one monic polynomial among them, which is often chosen).