Defining a Galois Field based on primitive element versus polynomial? Normally I see $GF(p^n)$ defined in terms of a reducing polynomial $P(x)$ of degree n, where the coefficients are elements of $GF(p)$. For example, $GF(2)[x]/\langle x^4+x+1 \rangle$ or $GF(2)[x]/\langle x^8+x^4+x^3+x+1 \rangle$. The number of primitive elements (multiplicative generators) for $GF(p^n) = ϕ(p^n-1)$ (where $ϕ$ is the totient function). For $GF(2^4)$, there are 8 generators, for $GF(2^8)$ there are 128 generators, and typically one of the generators is specified. If $P(x)$ is primitive, then one of the generators is $x$. There are algorithms to create a complete list of all generators for any $P(x)$ if needed.
I sometimes see an alternative based definition based on a polynomial of primitive elements | multiplicative generators. Let $A(\alpha)$ represent such a polynomial, where the coefficients are elements of $GF(p^n)$. For example, $A(\alpha) = ^4 +  + 1 = 0$. The actual wording is typically similar to: let $$ be a primitive element (multiplicative generator) of $GF(2^4)$ such that $^4 +  + 1 = 0$.
If $A(\alpha)$ is a primitive polynomial, its $n$ roots are generators for any $GF(p^n)$ . For example:
$A(\alpha) = ^4 +  + 1 = 0$ 
$GF(2)[x]/\langle x^4+x+1 \rangle$ : $(x)=\{x, x+1, x^2, x^2+1\}$.
$GF(2)[x]/\langle x^4+x^3+1 \rangle$ : $(x) = \{x^2+x, x^2+x+1, x^3+x^2, x^3+x^2+1\}$
$GF(2)[x]/\langle x^4+x^3+x^2+x+1 \rangle$ : $(x) = \{x^2+x, x^2+x+1, x^3+x, x^3+x+1\}$
If $A(\alpha)$ is not primitive, then the roots of $A(\alpha)$ will not be generators of $GF(p^n)$.
$A(\alpha)$ has $n$ roots, which is only a sub-set of the generators. For $GF(2^8)$, $A(\alpha)$ has 8 roots, while $GF(2^8)$ has 128 generators.
Since the second definition has limitations, what is the purpose of using the second definition?

Here is a link to an online document that uses the second definition, which is sufficient for determining minimum polynomials for BCH code, but then follows that with a table on page 6 that is specific to $GF(2)[x]/\langle x^4+x+1 \rangle$
https://web.ntpu.edu.tw/~yshan/BCH_code.pdf
 A: When you say:

What I was unaware of is defining $\alpha$ to be the sum of a primitive element plus the reducing polynomial, such as $\alpha = x + (x^4 + x + 1) = x^4 + 1$.

I believe that you are not reading the notation correctly. The notation $x + (x^4+x+1)$ here does not mean the sum of the polynomials $x$ and $x^4 + x + 1$. The $(x^4 + x + 1)$ in parentheses here means the ideal generated by $x^4 + x + 1$ in the ring $GF(2)[x]$, and $x + (x^4 + x + 1)$ means the coset of this ideal containing $x$. In other words, $x + (x^4 + x + 1)$ denotes the equivalence class of $x$ modulo $x^4 + x + 1$ in the quotient ring $GF(2)[x] / (x^4 + x + 1)$.
Granted, it would probably have been a little clearer to use a different symbol than ordinary parentheses for this purpose (for example, some people write it with angle brackets like $\langle x^4 + x + 1 \rangle$), but the parentheses notation is pretty standard in this context.
Answer to edited question:

I sometimes see an alternative based definition based on a primitive element, such as: let $\alpha$ be a primitive element of $GF(2^4)$ such that $\alpha^4+\alpha+1=0$.

I think the confusion here may be that there are two different uses of the term "primitive element" for an extension $K/F$ of fields. The first definition applies only to finite fields, and means a generator of the multiplicative group of the field. The second applies to any field extension, and refers to an element $\alpha \in K$ such that $K = F(\alpha)$, i.e., $\alpha$ generates $K$ over $F$ as a field. These are two totally different concepts.
One more point. In this statement that you quoted above:

Let $\alpha$ be a primitive element (multiplicative generator) of $GF(2^4)$ such that $\alpha^4+\alpha+1=0$.

we are still considering polynomials in $GF(2)[x]$ modulo $x^4 + x + 1$. A statement like the one above is really shorthand for 2 statements: The author wants to consider $GF(2^4)$ in the form $GF(2)[x] / \langle x^4 + x + 1 \rangle$, and is simultaneously claiming that $\alpha = x + \langle x^4 + x + 1 \rangle$ is a multiplicative generator of $GF(2)[x] / (x^4 + x + 1)$.  We have to check this claim (which is not an automatic fact and needs to be checked - not all choices of polynomials would give $\alpha$ as a multiplicative generator).
I don't understand when you keep saying that there are 4 vs 8 multiplicative generators. We are not saying that all the multiplicative generators satisfy $x^4 + x + 1 = 0$. In the statement above, we are only saying that $\alpha$ does. $GF(2^4)$ is unique up to isomorphism and has 8 multiplicative generators, no matter how you define it.
