You need to know: Field, field of rational numbers, isomorphic fields, vector space, vector space over a field, dimension of a vector space, matrix, determinant of a matrix.
Background: Number field is a field F that contains and has finite dimension n when considered as a vector space over . Number n is called the degree of F. A set of n elements of F is called basis of F if every can be written as with coefficients . Sum does not depend on the choice of basis e and is called trace of x. If, for every , all are integers, e is called integral basis of F. The determinant of an matrix with entries , , , does not depend on the choice of integral basis e and is called the discriminant of F. For , let denotes the number of non-isomorphic number fields of degree n with absolute value of the discriminant at most X.
The Theorem: On 8th September 2003, Jordan Ellenberg and Akshay Venkatesh submitted to arxiv a paper in which they proved the existence of constant depending only on n and absolute constant C, such that inequality holds for all and all .
Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is a conjecture that grows as linear function of X for every fixed n, but, before 2003, this was known only for (in later works – see here and here – Bhargava proved it for and ). For general n, the best upper bound was . The Theorem proves a bound which is significantly better for large n.
Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.2 of this book for an accessible description of the Theorem.
Theorems of the 21st century 
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