query: Prove that given two elements $a,b$ of relatively prime degree in a field extension $K$, $F(a) \cap F(b) = F$
positive_passages.docid: 147069
positive_passages.text: Intersection of field extensions
negative_passages.docid: 26766
negative_passages.text: Proving : $A \cap (B-C) = (A \cap B) - (A \cap C)$
negative_passages.docid: 92677
negative_passages.text: Infinite algebraic extension of a finite field
negative_passages.docid: 10107
negative_passages.text: Prove $A\cup (A\cap B) = A$
negative_passages.docid: 5360
negative_passages.text: Ideal contained in a finite union of prime ideals
negative_passages.docid: 68918
negative_passages.text: Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$
negative_passages.docid: 19511
negative_passages.text: Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed?
negative_passages.docid: 111114
negative_passages.text: Prove that an ideal in $k[x,y]$ is a prime ideal
negative_passages.docid: 111129
negative_passages.text: Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
negative_passages.docid: 41477
negative_passages.text: Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
negative_passages.docid: 41815
negative_passages.text: Galois extension of intersection of fields
negative_passages.docid: 138069
negative_passages.text: Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
negative_passages.docid: 53551
negative_passages.text: $f(A\cap B)=f(A)\cap f(B)$ $\iff$ $f$ is injective.
negative_passages.docid: 26999
negative_passages.text: How is the degree of the minimal polynomial related to the degree of a field extension?
negative_passages.docid: 86877
negative_passages.text: Let $F$ be a Galois extension over $\mathbb{Q}$ with $[F:\mathbb{Q}]=2^n$, then all elements in $F$ are constructible
negative_passages.docid: 79450
negative_passages.text: For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$
negative_passages.docid: 27250
negative_passages.text: Greatest common divisor of two relatively primes