Given a field K, a quadratic extension field L is an extension of K that can be generated from K by adding a root of a quadratic polynomial with coefficients in K. This paper shows how ACL2(r) can be used to reason about chains of quadratic extension fields Q = K_0, K_1, K_2, ..., where each K_i+1 is a quadratic extension field of K_i. Moreover, we show that some specific numbers, such as the cube root of 2 and the cosine of pi/9, cannot belong to any of the K_i, simply because of the structure of quadratic extension fields. In particular, this is used to show that the cube root of 2 and cosine of pi/9 are not rational.

中文翻译：

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ACL2 中的二次扩展

给定域 K，二次扩展域 L 是 K 的扩展，可以通过添加系数在 K 中的二次多项式的根从 K 生成。 本文展示了如何使用 ACL2(r) 来推理链二次扩展域 Q = K_0, K_1, K_2, ...，其中每个 K_i+1 是 K_i 的二次扩展域。此外，我们证明了一些特定的数字，例如 2 的立方根和 pi/9 的余弦，不能属于任何 K_i，仅仅是因为二次扩展域的结构。特别是，这用于表明 2 的立方根和 pi/9 的余弦是不合理的。