| Let N be a finitely-dimensional commutative nilpotent algebra over a field k of char-acteristic p>0. A conjecture of Eggert (1971) says that dim N≥p dim Np, where Np is the subalgebra of N generated by elementsχp,χ∈N. Eggert proved his conjecture only when dim Np≤2. In 1976, Bautista proved it when dim Np=3, a shorter proof of which is given by Stack in 1998 whenκis a perfect field. Amberg and Kazarin (2001) proved the conjecture for the case dim Np≤4. Other results are presented by McLean in 2004 and 2006, respectively. He showed that this conjecture is true if the algebra N is graded and at least one of the following conditions is fulfilled:(i) s= 2,3; (ii) p=2, and s=4;(iii) p=3, and s=3;(iv)n<3p, and 3≤s-1≤p. In a paper concerning Eggert's con-jecture appeared in 2002, Hammoudi claimed he proved the conjecture. However, as pointed by Amberg and Kazarin, Hammoudi's proof has a gap. A counterexample to Hammoudi's method was also provided by McLean. In 2010, Miroslav showed that Eggert's conjecture is true if the subalgebra Np has at most two generators.This thesis is a survey on the Eggert's conjecture on finitely dimensional commutative nilpotent algebras. The first section mainly expounds the development of Eggert's conjec-ture. In the section 2, we introduce the conjecture posed by Eggert in 1971 and some results due to several authors. Then the proof with gaps due to Hammoudi is sketched. In 2005, Hammoudi corrects his proof and claim that the corrected proof proves only a particular case of Eggert's conjecture and dose not solve Eggert's conjecture completely. Eggert's conjec-ture in general remains open now. Next, we introduce a non-commutative version of Eggert's conjecture posed by Stack. Stack proved that the relation dim N≥p dim Np holds for a noncommutative algebra N as well as under the assumption that the dimension of Np does not exceed two. And Stack conjectured that the Eggert conjecture is true for any noncommu-tative nilpotent algebra. However, counterexamples given by the others shows that Stack's conjecture need not be true. In Section 5, Eggert's conjecture on graded algebras are dis-cussed briefly. McLean shows that Eggert's conjecture is true in certain commutative graded cases. He conjectures that:H(j)≥h(s-1)for all j such that p(s-2) |
