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Is there a fundamental difference between transcendental elements and variables?

If I look at the polynomial ring $K[X]$ over a field $K$, I see no difference with the ring extension $K[\alpha]$ where $\alpha$ is transcendental over $K$, we should even have $K[X]\cong K[\alpha]$. Also, this is often used to argue that, for instance, $$ f(T)=T^2+XT+XY\in K(X,Y)[T] $$ is irreducible because it is $X$-Eisenstein. Can somebody explain to me the mathematical difference between these two terms?

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  • $\begingroup$ There's really no difference up to an isomorphism. $\endgroup$ – Bernard May 21 '18 at 12:41
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If $\alpha$ is transcendental over $K$ and $T$ is an indeterminate (or variable, if you prefer), then $K[\alpha]$ is isomorphic to $K[T]$.

The main difference is just that you don't need to mention an extension field where to pick $\alpha$ from. If you want to consider several (maybe infinitely many) variables, it's best to use indeterminates rather than transcendental elements. But in the end, this amounts to the same thing: if $S$ is a set of indeterminates over $K$ and $K(S)$ denotes the field of fractions of $K[S]$ (polynomials in the given indeterminates), then every element of $S$ is transcendental over $K$ (and they are algebraically independent).

If you want to talk about a transcendency degree $2$ field extension of $\mathbb{Q}$, you cannot generally pick any two transcendental real numbers, because they could be algebraically dependent (think to $e$ and $e^2$). On the other hand, $K(X,Y)$, where $X$ and $Y$ are (distinct) indeterminates over $K$, is a transcendency degree $2$ field extension of $K$.

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