Definition talk:Degree of Polynomial

It seems to me that the main definition is inconsistent. For the representation of $f = \sum a_i x^i$ need not be unique. Consider $x = 0$.

I'm fine with the extra generality obtained by considering the indeterminate to reside in a higher field, instead of being an "indeterminate", but then this needs to be formalised and the degree needs to be defined as an appropriate infimum. — Lord_Farin (talk) 11:46, 13 May 2017 (EDT)

Agreed. It seems like this is how the problem arose:

• Book:Thomas A. Whitelaw/An Introduction to Abstract Algebra defines "a polynomial in $x$ over $D$" , with $D$ an integral domain and $x\in R\supset D$, $R$ a ring, $x$ transcendental over $D$.
• He then defines the degree of such a polynomial. This is fine, because of the conditions imposed on $x$ and $D$.
• $\mathsf{Pr} \infty \mathsf{fWiki}$ adopts this definition of polynomial and degree, which is fine.
• Other sources are consulted to back up a definition of polynomials over general rings. They define polynomials in an abstract way; e.g. as sequences: Book:B. Hartley/Rings, Modules and Linear Algebra. Cited at Definition:Polynomial Ring over Sequences
• The definition of degree in those sources is mixed up with the one in Whitelaw: Hartley & Hawkes definition 3.6 of degree is not the one at Definition:Degree of Polynomial over Ring (I checked)

--barto (talk) 15:43, 12 July 2017 (EDT)