# Which definition of dimension came first?

In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, or, the Krull dimension of $R$. A few minutes later, we were given the definition in terms of transcendence degree: the dimension of $X$, if $X$ is integral, is the transcendence degree of $K(R)$ over $k$. I have also seen dimension defined in terms of the degree of Hilbert polynomials (Theorem C, page 225 in Eisenbud).

My question: which of these came first and why? Why were the other definitions developed? What did they allow us to do more easily or effectively that previous definitions did not?

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Search this - encyclopediaofmath.org/index.php/Commutative_algebra - page for the word "dimension" - it containts a lot of interesting commutative algebra history. In particular, it seems like dimension were originally defined as the transcendence degree of $K(R)/k$. –  Fredrik Meyer Sep 30 '12 at 15:30
I think you're right: "Beginning with Kronecker and Lasker's dimension, defined then as the transcendence degree of the quotient ring corresponding to a prime ideal in a ring of polynomials, the modern combinatorial definition of dimension was subsequently proposed by W. Krull." –  Derek Allums Sep 30 '12 at 15:39
Actually, if you have Eisenbud at hand, you can read chapter 8, which gives a short history of dimension in algebraic geometry. –  Andrew Sep 30 '12 at 16:42
@Andrew Well this is embarrassing: I cited Eisenbud in my question but didn't think to turn back a few pages. Thanks. If you convert this to an answer I'll accept it. –  Derek Allums Oct 1 '12 at 17:36