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# 8.3: Dominios euclidianos

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Crear un campo de fracciones es una forma de resolver definitivamente los problemas de división en un dominio integral: Componer fracciones para tener una inversa para cada elemento distinto de cero. Pero hay (a veces) otra manera de definir la división sin recurrir a introducir nuevos elementos al campo, familiares a partir de los enteros: definir la división usando un 'cociente' y un 'resto'.

Por ejemplo, entre los enteros podemos escribir$$25 = 8\cdot 3+1$$; then $$25/\mathord 3$$ has a quotient $$8$$ and remainder $$1$$. Generally, to find $$n/\mathord m$$, we write $$n= qm+r$$, where $$0<r<|m|$$. Then $$q$$ is the quotient and $$r$$ is the remainder.

Podemos hacer algo similar con polinomios: Dados dos polinomios$$f$$ and $$g$$, we can divide $$f$$ by $$g$$ and uniquely write $$f=Qg+R$$, where $$Q$$ is a polynomial and $$R$$ is a polynomial of lower degree than $$g$$.

Por ejemplo, tome$$f=2x^5+3x^2+x+3$$ and $$g=x^2+1$$, we can apply the polynomial long division algorithm and get $$f= (2x^3-2x+3)g -x$$. Here $$2x^3-2x+3$$ is the whole part and $$-x$$ is the remainder.

Tanto en la división entera como en la división polinómica, el ingrediente clave es una forma de ordenar los elementos del anillo: en los enteros, ordenamos por el orden habitual de los enteros, y con polinomios ordenamos por el grado del polinomio.

##### Definición 8.2.0: Norma en un Anillo

Una norma en un anillo$$R$$ is a function $$n: R\rightarrow \mathbb{Z}_{\geq 0}$$ with $$n(0)=0$$. A positive norm has $$n(r)>0$$ for all $$r\neq 0$$.

Cualquier anillo dado puede tener muchas normas diferentes. La norma sobre los enteros es simplemente el valor absoluto del entero; es una norma positiva. La norma en polinomios es el grado del polinomio.

##### Definición 8.2.1: Dominio euclidiano

Un dominio euclidiano es un dominio integral$$R$$ with a norm $$n$$ such that for any $$a, b\in R$$, there exist $$q,r$$ such that $$a=q\cdot b + r$$ with $$n(r)<n(b)$$. The element $$q$$ is called the quotient and $$r$$ is the remainder.

Un dominio euclidiano tiene entonces el mismo tipo de solución parcial a la cuestión de la división que tenemos en los enteros.

De hecho, los dominios euclidianos tienen además un algoritmo euclidiano para encontrar un divisor común de dos elementos. El algoritmo euclidiano se realiza comenzando con dos elementos$$f$$ and $$g$$ for which we wish to find a common divisor. Dividing $$f$$ by $$g$$ gives a quotient $$q_0$$ and a remainder $$r_0$$. We then divide $$g$$ by $$r_0$$ and obtain a new quotinet $$q_1$$ and a new remainder, $$r_1$$. We then repeat this process to get quotients $$q_2, q_3, \ldots q_k$$ and remainders $$r_2, r_3, \ldots r_k$$. Each remainder has smaller norm than the previous, so this process must eventually terminate with some $$r_k=0$$.

El cociente final,$$q_k$$ divides both $$g$$ and $$f$$: You can see this by writing $$f=q_0g+r_0$$, and then expanding $$r_0$$: $$f=q_0(q_1r_0 + r_1)+r_0$$. If we imagine the process ending at this point, so that $$r_1=0$$, we then have $$r_0$$ divides both $$f$$ and $$g$$. On the other hand, if the process doesn't terminate, we can expand $$r_0=q_2r_1+r_2$$. Then $$f=q_0(q_1(q_2r_1+r_2) + r_1)+q_2r_1+r_2$$. If the process terminates, then $$r_2=0$$, and $$r_1$$ divides every term, and thus divides $$f$$ and $$g$$. If the process doesn't terminate, we repeat the same basic argument.

(TODO: Ejemplos en Z y Z [x])

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This page titled 8.3: Dominios euclidianos is shared under a not declared license and was authored, remixed, and/or curated by Tom Denton.