Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider: "Let $K$ be a field. Then every polynomial $p\in K[X]$ is transcendent over $K$." My question are:

1) How can polynomial be transcendent over something ? I thought that definition applied only for elements of $K$...

2) How can I show the above ? Do I have to find a polynomial "whose coefficients are polynomials" such that $p$ is the root of this greater polynomial ?

share|cite|improve this question
Surely you mean every polynomial of degree $\ge 1$. – Robert Israel Oct 30 '12 at 21:42
Elements of $K$ can't possibly be transcendental over $K$ --- only an element of an extension of $K$ (such as, for example, a polynomial) can be transcendental over $K$. – Gerry Myerson Oct 30 '12 at 21:43
up vote 1 down vote accepted

Just think about for example the fact that $\mathbb Q(\pi)$ and $\mathbb Q[X]$ are isomorphic. They're both transcendental extensions of $\mathbb Q$.

share|cite|improve this answer

Hint: if $p(X)$ is a polynomial of degree $d$ with coefficients in $K$, and $Q$ is a polynomial of degree $q$ with coefficients in $K$, then $Q(p(X))$ is a polynomial of degree ...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.