# Adjoining a transcendental element to an extension and a base field

Let $$L/K$$ be a finite field extension. Let $$x$$ be transcendental over $$L$$. Why is it true that $$[L(x):K(x)]=[L:K]?$$ Any reference?

• You can write any element of $L(x)$ as $h(x)=\frac{\sum_jp_j(x)r_j}{\sum_jq_j(x)r_j}$, where $r_1,...,r_n$ is a basis of $L$ over $K$ and $p_i,q_j\in K[x]$. Then, multiplying and dividing by $\prod_i(\sum_j q_j(x)r_{j,i})$, where the product ranges over all conjugates of the $r_i$ gives a denominator that is in $L[x]$. Therefore, $h$ gets written as $\sum_i h_j(x)r_i$, where $h(x)\in K(x)$. This shows that $L(x)$ is generated by $r_1,...,r_n$ over $K(x)$. They are also independent over $K(x)$ since they are independent over $K$ and equality between polynomials is termwise. – user647486 Apr 26 at 17:18
• Above there is a missing subscript. It should read where $h_i\in K(x)$. – user647486 Apr 26 at 17:28