Lower bound on the dimension of the field of rational functions

Let $𝕂$ be a field and $𝕂 (𝑥)$ be its field of rational functions in indeterminate $𝑥$ . Then the $𝕂$ -dimension of $𝕂 (𝑥)$ is at least the cardinality of $𝕂$ . #m/thm/ring

| 𝕂 | \leq d i m 𝕂 𝕂 (𝑥)

In particular, the following set linearly independent:

𝑆 = {1 𝑥 - 𝜆 : 𝜆 \in 𝕂}

Proof

Let

𝑓 𝑛 (𝑥) = 𝑝 𝑛 (𝑥) 𝑞 𝑛 (𝑥) = 𝑛 \sum 𝑖 = 1 1 𝑥 - 𝜆 𝑖

where

𝑞 𝑛 (𝑥) = 𝑛 \prod 𝑖 = 1 (𝑥 - 𝜆 𝑖)

then

𝑓 𝑛 + 1 (𝑥) = 𝑝 𝑛 (𝑥) 𝑞 𝑛 (𝑥) + 1 𝑥 - 𝜆 𝑛 + 1 = 𝑝 𝑛 (𝑥) (𝑥 - 𝜆 𝑛 + 1) + 𝑞 𝑛 (𝑥) 𝑞 𝑛 (𝑥) (𝑥 - 𝜆 𝑛 + 1)

which is zero iff $𝑞 𝑛 (𝑥) = - (𝑥 - 𝜆 𝑛 + 1) 𝑝 𝑛 (𝑥)$ . But this is impossible since $(𝑥 - 𝜆 𝑛 + 1) ∤ 𝑞 𝑛 (𝑥)$ .

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