ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 22 Jun 2012 17:42:23 -0500How does one tell if a number field contains a certain element?http://ask.sagemath.org/question/9094/how-does-one-tell-if-a-number-field-contains-a-certain-element/Suppose I have a number field K, in SAGE how would can see if something like \sqrt[4]{29} is contained in this number field?Tue, 19 Jun 2012 01:32:47 -0500http://ask.sagemath.org/question/9094/how-does-one-tell-if-a-number-field-contains-a-certain-element/Answer by benjaminfjones for <p>Suppose I have a number field K, in SAGE how would can see if something like \sqrt[4]{29} is contained in this number field?</p>
http://ask.sagemath.org/question/9094/how-does-one-tell-if-a-number-field-contains-a-certain-element/?answer=13750#post-id-13750I think your question is not well-defined. If you had a number field $K$ along with an embedding $K \to \mathbb{C}$ then the question makes sense. When you construct a number field in Sage I believe it is an abstract field since there may be many different embeddings of $K$ into $\mathbb{C}$.
Now, if your field $K$ is the splitting field of an irreducible polynomial $f$ over $\mathbb{Q}$ then you could test if $f(\sqrt[4]{29})$ is zero or not.Fri, 22 Jun 2012 17:42:23 -0500http://ask.sagemath.org/question/9094/how-does-one-tell-if-a-number-field-contains-a-certain-element/?answer=13750#post-id-13750