$Z = \{x \in K \mid \sigma_1(x) = \cdots = \sigma_n(x)\}$ is a subfield of K with $[K : Z] \geq n$.

Question

Let $L/K$ be a field extension and let $\sigma_1, \cdots, \sigma_n : K \to L $ be pairwise different nonzero field homomorphisms. Show that $Z = \{x \in K \mid \sigma_1(x) = \cdots = \sigma_n(x)\}$ is a subfield of K with $[K : Z] \geq n$.

I have proved that $Z$ is a subfield. Help needed in the 2nd part. I guess to use contradiction.

I also have a confusion in understanding the concept of pairwise different nonzero field homomorphisms.

Answer 1

Pick a basis $ \beta_1, \beta_2, \ldots, \beta_m $ for the extension $ K/Z $. Consider the matrix $ M = (\sigma_i(\beta_j)) $, with entries in $ L $. The rows of this matrix are linearly independent over $ L $: indeed, a linear dependence relation would take the form

$$ c_1 \sigma_1(\beta_j) + c_2 \sigma_2(\beta_j) + \ldots + c_n \sigma_n(\beta_j) = 0 $$

for all $ \beta_j $, and by identifying $ Z $ with its image in $ L $ and taking $ Z $-linear combinations, we would obtain a linear dependence relation between distinct characters $ K^{\times} \to L $ - contradicting linear independence of characters.

Since the rows of $ M $ are linearly independent, it follows by linear algebra that $ M $ must have at least as many columns as rows. $ M $ has $ m $ columns and $ n $ rows, thus $ m \geq n $. $ m = [K : Z] $ by definition, so the result follows.