Aula 22 - Álgebra II

[Conclusão da aula anterior: exemplos de polinómios resolúveis e polinómios não resolúveis]

4. Corpos finitos

Neste capítulo final vamos estudar as propriedades fundamentais dos corpos finitos e descrever algumas das suas muitas aplicações (à teoria dos códigos, teoria dos números e teoria matemática dos jogos).

O corpo 𝔽_{p} = (ℤ_{p}, ⊕_{p}, ⊗_{p}) dos inteiros módulo p (p primo) é, evidentemente, o exemplo mais familiar de corpo finito. Muitas das suas propriedades generalizamse aos corpos finitos arbitrários. Os corpos 𝔽_{p} representam um papel muito importante na teoria dos corpos pois, como vimos, todo o corpo de característica p contém uma cópia isomorfa de 𝔽_{p} (como seu subcorpo primo) e pode então ser visto como uma extensão de 𝔽_{p}. Esta observação, conjuntamente com o facto óbvio de que todo o corpo finito tem característica finita (prima), é fundamental para a classificação dos corpos finitos.

Além dos corpos 𝔽_{p}, de ordem prima p, já encontrámos (na Aula 17) outros exemplos de corpos finitos: um corpo de ordem 4 = 2^{2}, definido pelas tabelas

\begin{matrix}+ & 0 & 1 & α & β \\ 0 & 0 & 1 & α & β \\ 1 & 1 & 0 & β & α \\ α & α & β & 0 & 1 \\ β & β & α & 1 & 0 \end{matrix}  \begin{matrix}· & 0 & 1 & α & β \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & α & β \\ α & 0 & α & β & 1 \\ β & 0 & β & 1 & α \end{matrix}

e um corpo de ordem 16 = 2^{4}. Haverá algum corpo de ordem 6? Veremos em seguida que não, ao provarmos que a ordem de qualquer corpo finito é necessariamente da forma p^{n} para algum primo p e algum natural n, e que, para cada número dessa forma existe, a menos de isomorfismo, exactamente um corpo com esse número de elementos.

Antes de avançarmos para a prova desses resultados que permitem classificar os corpos finitos, vejamos uma aplicação do corpo com 4 elementos acima referido, que se pode encontrar em [N. de Bruijn, A solitaire game and its relation to a finite field, J. Recreational Math. 5 (1972) 133].

O jogo do solitário é jogado num tabuleiro como a figura representa

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∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘

Inicialmente, em cada buraco, com excepção do central, coloca-se uma bola (32 bolas no total):

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∘ • • • • • • • • • • • • •  • • • • • • • • • • • • •  • • • • • •

O jogo desenrola-se movimentando uma bola por cima de outra adjacente (na vertical ou na horizontal) para um buraco vazio; a bola sobre a qual se saltou é então removida do jogo. O objectivo do jogador é chegar a uma situação em que só reste uma bola no tabuleiro.

                        [Frequentemente, esta última bola acaba por
                             ficar no buraco central; experimente!]

Poderá ficar noutro buraco que não o central?

Se jogarmos algumas vezes observaremos que sim, mas também nos convenceremos que talvez não possa ocupar qualquer posição.

Quais são as posições possíveis?

A ideia de de Bruijn é usar o corpo acima referido para determinar tais posições. Para isso, consideremos os buracos do tabuleiro referenciados por pares de inteiros (i, j), com o buraco central em (0, 0):

∘ • • • • • • • • • • • • •  • • • • • • •  • • • • • •  • • • • • •  ^{(0, 0)}^{(0, 1)}^{(0, 2)}^{(0, 3)}^{(0, −1)}^{(0, −2)}^{(0, −3)}^{(−1, 0)}^{(−1, 1)}^{(−1, 2)}^{(−1, 3)}^{(−1, −1)}^{(−1, −2)}^{(−1, −3)}  ^{(1, 0)}^{(1, 1)}^{(1, 2)}^{(1, 3)}^{(1, −1)}^{(1, −2)}^{(1, −3)}  ^{(2, 0)}^{(2, 1)}^{(2, −1)}^{(−2, 0)}^{(−2, 1)}^{(−2, −1)}  ^{(3, 0)}^{(3, 1)}^{(3, −1)}^{(−3, 0)}^{(−3, 1)}^{(−3, −1)}

Definamos, para cada conjunto X de bolas colocadas no tabuleiro, os números

A(X) = ∑_{(i,j)∈X} α^{i+j}, B(X) = ∑_{(i,j)∈X} α^{i−j}.

Por exemplo, para a posição inicial X_{1} do jogo, é fácil de ver (observe a figura abaixo) que

A(X_{1}) = B(X_{1}) = 2α^{4} + 4α^{3} + 5α^{2} + 4α^{1} + 2α^{0} + 4α^{−1} + 5α^{−2} + 4α^{−3} + 2α^{−4}
= 0 + 0 + 5β + 0 + 0 + 0 + 5α + 0 + 0
= α + β = 1.

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Cada jogada, que transforma um conjunto X de bolas no tabuleiro num conjunto Y, é necessariamente de um dos quatro tipos seguintes:

X  Y • • ∘⋮⋮ · · · · · · ⇝  ∘ ∘ •⋮⋮ · · · · · ·  ∘ • •⋮⋮ · · · · · · ⇝  • ∘ ∘⋮⋮ · · · · · · X  Y

X  Y • • ∘ ⋮⋮ ⋯  ⋯ ⇝  ∘ ∘ • ⋮⋮ ⋯  ⋯  ∘ • • ⋮⋮ ⋯  ⋯ ⇝  • ∘ ∘ ⋮⋮ ⋯  ⋯ X  Y

É fácil de ver que, em qualquer um desses tipos de jogada, se tem A(Y) = A(X) e B(Y) = B(X). Por exemplo, no primeiro tipo, se supusermos que a bola a movimentar está inicialmente na posição (i, j) (e portanto, após a jogada, vai ficar na posição (i, j + 2)), então

A(X) − A(Y) = α^{i+j} + α^{i+j+1} − α^{i+j+2} = α^{i+j}(1 + α + α^{2}) = 0,

B(X) − B(Y) = α^{i−j} + α^{i−j−1} − α^{i−j−2} = α^{i−j}(1 + β + β^{2}) = 0.

Portanto, o par (A(X), B(X)) é invariante ao longo do jogo.

Assim, se o jogo terminar com uma só bola no tabuleiro, na posição (i, j), teremos necessariamente A({(i, j)}) = 1 e B({(i, j)}) = 1, isto é, α^{i+j} = 1 e α^{i−j} = 1. Como as sucessivas potências de α são

α^{−4} = β, α^{−3} = 1 , α^{−2} = α, α^{−1} = β, α^{0} = 1 , α^{1} = α, α^{2} = β, α^{3} = 1 , α^{4} = α,

a posição (i, j) da bola final terá que satisfazer i+j ∈ {−3, 0, 3} e i−j ∈ {−3, 0, 3}:

• ∘ ∘ • ∘ ∘ • ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  • ∘ ∘ • ∘ ∘  ^{i + j = 3}^{i − j = 3}^{i + j = 0}^{i − j = 0}^{i + j = −3} ^{i − j = 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Em conclusão, as únicas posições finais possíveis são (−3, 0), (0, −3), (0, 0), (0, 3) e (3, 0):

• ∘ ∘ • ∘ ∘ • ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  • ∘ ∘ • ∘ ∘

Por experimentação, é possível concluir que todas elas podem ser, de facto, obtidas.

                       [Por simetria, basta mostrar que se consegue
                                   atingir as posiç~ oes (0, 0) e (3, 0)]