假設F是一個特徵數為2的體而ϕ:M_n(F)→M_n(F)是一個Jordan同構。在本篇論文中，我們將證明當ϕ是F-線性且n=2時，則ϕ一定是同構或反同構。我們同時也舉出反例來說明，當ϕ少了F-線性的條件後，對於任意一個大於或等於2的整數n，此結果是不成立的。

Let F be a field with char(F) = 2, and let ϕ : M_n(F) → M_n(F) be a Jordan isomorphism.
We prove that if ϕ is F-linear and n = 2, then ϕ is either an isomorphism or an anti-isomorphism,
and it is not true if ϕ is not F-linear for all n ≥ 2.

論文審定書. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . i
中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2
2.1 Jordan isomorphisms on prime rings of characteristic not 2 . . . . . . . . . . . . .2
2.2 F-linear Jordan isomorphisms of Mn(F), n > 2 with characteristic 2. . . . . . . 7
3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . .. 9
3.1 F-linear Jordan isomorphisms of Mn(F), n = 2 with characteristic 2. . . . . . . 9
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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