We study which fields F can be represented as finite sums of proper subfields. We prove that for any $$n \ge 2$$ n ≥ 2 every field F of infinite transcendence degree over its prime subfield can be represented as an unshortenable sum of n subfields, and every rational function field $$F = K(x_1, \ldots , x_n)$$ F = K ( x 1 , … , x n ) can be represented as an unshortenable sum of $$n + 1$$ n + 1 subfields. We also show that no subfield of the algebraic closure of a finite field is a finite sum of proper subfields, and no finite extension of the field $${\mathbb {Q}}$$ Q of rationals can be decomposed into a sum of two proper subfields.

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关于域作为适当子域的有限和的表示

我们研究哪些域 F 可以表示为适当子域的有限和。我们证明，对于任何 $$n \ge 2$$ n ≥ 2，其质子域上无限超越度的域 F 都可以表示为 n 个子域的不可缩短的和，并且每个有理函数域 $$F = K(x_1 , \ldots , x_n)$$ F = K ( x 1 , … , xn ) 可以表示为不可缩短的 $$n + 1$$ n + 1 个子字段的总和。我们还表明，有限域的代数闭包的任何子域都不是真子域的有限和，并且域的有限扩展 $${\mathbb {Q}}$$ Q 的有理数不能分解为两个适当的子字段。