Talk:Algebraic extension

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I believe Extension (Algebra) should be merged into the main topic Algebraic extension. I know that an extension and an algebraic extension are different things, but it would help those reading algebraic extension, if they knew what an extension was first, and simply linking it at the end of the piddly Extension_(Algebra) is more of a "Would you like to know more?" than a "See also." IMHO. Sim 01:58, 1 April 2006 (UTC)[reply]

"If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is a field."

Is this actually a field? What's the multiplicative inverse of a?

Suppose that ${\displaystyle f(X)=X^{n}+b_{n-1}X^{n-1}+\ldots +b_{0}}$ is the minimal polynomial for a over K (we can safely assume that it's monic). Then ${\displaystyle a^{n}+b_{n-1}a^{n-1}+\ldots +b_{0}=0}$, so if you move the constant term over and factor out an a, you get ${\displaystyle a(a^{n-1}+b_{n-1}a^{n-2}+\ldots +b_{1})=-b_{0}}$. Since K is a field, ${\displaystyle -b_{0}}$ has a multiplicative inverse, so in particular, ${\displaystyle a(a^{n-1}+b_{n-1}a^{n-2}+\ldots +b_{1})(-b_{0})^{-1}=1}$. Thus that big ugly thing in the parentheses and that negative-b-naught-inverse is a's multiplicative inverse. HTH - if you have questions, feel free to stick them on my talk page. Druiffic (talk) 07:36, 21 February 2009 (UTC)[reply]

The term "sub K-algebra" seems stilted and clumsy to me - is "K-sub-algebra" bad? It sounds more natural. Druiffic (talk) 07:36, 21 February 2009 (UTC)[reply]

The article states: "Q[π] and Q[e] are fields but π and e are transcendental over Q." But Q[π] and Q[e] are isomorphic to the ring of polynomials Q[x], hence they are not fields. Danneks (talk) 07:59, 23 February 2023 (UTC)[reply]

 Fixed. D.Lazard (talk) 09:16, 23 February 2023 (UTC)[reply]