query: How to prove that algebraic numbers form a field?
positive_passages.docid: 24551
positive_passages.text: How to prove that the sum and product of two algebraic numbers is algebraic?
negative_passages.docid: 146047
negative_passages.text: Proving that $\cos(2\pi/n)$ is algebraic
negative_passages.docid: 8087
negative_passages.text: Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer
negative_passages.docid: 99516
negative_passages.text: Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing
negative_passages.docid: 73688
negative_passages.text: Can I use Gödel numbers to make a proof?
negative_passages.docid: 112556
negative_passages.text: When is $F(\sqrt c)$ a field?
negative_passages.docid: 144245
negative_passages.text: Shorter proof of $R/I$ is a field if and only if $I$ is maximal
negative_passages.docid: 16384
negative_passages.text: Proving a result in a given field
negative_passages.docid: 126427
negative_passages.text: Proving $\sqrt{2}+\sqrt[3]{3}$ is algebraic.
negative_passages.docid: 16163
negative_passages.text: Show $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a simple extension field of $\mathbb{Q}$.
negative_passages.docid: 89271
negative_passages.text: If $F$ is a field, then $F[x]$ is a principal ideal domain proof question
negative_passages.docid: 14055
negative_passages.text: How to show that a finite commutative ring without zero divisors is a field?
negative_passages.docid: 6160
negative_passages.text: How to prove a number $a$ is quadratic residue modulo $n$?
negative_passages.docid: 16589
negative_passages.text: Field of fractions of integral extension is an algebraic extension
negative_passages.docid: 13983
negative_passages.text: Show that If $R[X]$ is Euclidean domain then $R$ is a field
negative_passages.docid: 63280
negative_passages.text: Show that the set $\mathbb{Q}[\sqrt{2}] = \{a + b \sqrt{2} \mid a, b \in \mathbb{Q}\}$ is a field with the usual multiplication and addition.
negative_passages.docid: 142530
negative_passages.text: Let K be a field, and $I=(XY,(X-Y)Z)⊆K[X,Y,Z]$. Prove that $√I=(XY,XZ,YZ)$.