query: Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?
positive_passages.docid: 53701
positive_passages.text: Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$
negative_passages.docid: 104300
negative_passages.text: Help justifying that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field over $\mathbb Q$.
negative_passages.docid: 119832
negative_passages.text: Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$
negative_passages.docid: 6855
negative_passages.text: Subfields of finite fields
negative_passages.docid: 38549
negative_passages.text: $\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ iff $n=m$
negative_passages.docid: 154318
negative_passages.text: Can a field be isomorphic to its subfield?
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: 42810
negative_passages.text: Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
negative_passages.docid: 122083
negative_passages.text: Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$
negative_passages.docid: 130826
negative_passages.text: Computing class group of $\mathbb Q(\sqrt{6})$
negative_passages.docid: 115976
negative_passages.text: $\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?
negative_passages.docid: 99516
negative_passages.text: Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing
negative_passages.docid: 112909
negative_passages.text: is a number field by definition a subfield of $ \mathbb C $?
negative_passages.docid: 137711
negative_passages.text: Alternative proof that $[\Bbb{Q}(\zeta_n) : \Bbb{Q} ]= \varphi(n)$ uses circular reasoning?
negative_passages.docid: 127198
negative_passages.text: What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?
negative_passages.docid: 8977
negative_passages.text: Showing $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible
negative_passages.docid: 58009
negative_passages.text: How to show that $\mathbb Q(\sqrt 2)$ is not field isomorphic to $\mathbb Q(\sqrt 3).$