‘Is conjugate to over F’ is an equivalence relation

Let F be a field and let E be an algebraic extension of F. Prove that the relation \sim defined by \alpha \sim \beta if and only if \alpha and \beta are conjugate over F is an equivalence.

Recall that \alpha and \beta are called conjugate precisely when they have the same minimal polynomial over F. We can map E to the set S of all minimal polynomials of elements of E by sending each element to its minimal polynomial; call this map f. (Note that f is well-defined since minimal polynomials are unique.) Now \alpha \sim \beta precisely when f(\alpha) = f(\beta). Thus it is clear that \sim is an equivalence relation.

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: