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The Abel Prize is an international award for outstanding work in mathematics, named after Niels Henrik Abel. It was first awarded in 2003 to Jean-Pierre Serre. The Norwegian Academy of Science and Letters administers the selection.^[ref]

The mathematical work behind an award is more informative than the honour alone. The 2009 award recognized Mikhail Gromov's contributions to geometry and topology, including global geometric questions whose behaviour cannot be read from local coordinates alone.^[ref] The 2021 award to László Lovász and Avi Wigderson recognized contributions connecting discrete mathematics and theoretical computer science.^[ref] The prize's scope therefore spans several mathematical traditions.

The Abel-Ruffini theorem states that there is no formula using finitely many field operations and extraction of radicals that solves the general quintic equation. The same impossibility holds for the general polynomial equation of any degree of a polynomial at least five. It does not say that every individual quintic equation is insoluble by radicals.

In Galois theory, a characteristic-zero polynomial is solvable by radicals exactly when its Galois group is a solvable group. The general degree-$n$ polynomial has symmetric group $S_n$, which is not solvable for $n\ge 5$. In contrast, $x^5-2=0$ has roots $\sqrt[5]{2}{\zeta }_5^j$, so it is solvable by radicals. Niels Henrik Abel's 1824 memoir established the general quintic impossibility; its title states precisely that goal.^[ref]

A German mathematician who made major contributions to elliptic functions, number theory and Hamiltonian mechanics. His 1829 work on elliptic functions developed a theory using theta functions, during the same period in which Niels Henrik Abel independently advanced the subject.

He also investigated first-order partial differential equations arising in dynamics and wrote extensively on determinants. The Jacobian determinant bears his name: it measures local volume scaling for a differentiable change of variables. This connects his work to both coordinate transformations in multivariable calculus and the study of dynamical equations. His Königsberg teaching helped establish research seminars as a way to train mathematicians. ^[ref]

Homepage: mathshistory.st-andrews.ac.uk/Biographies/Jacobi/