query: $[EK:F]\leq [E:F][K:F]$
positive_passages.docid: 113411
positive_passages.text: If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$?
negative_passages.docid: 48291
negative_passages.text: Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
negative_passages.docid: 147627
negative_passages.text: Relation between condition numbers $\kappa(A^T A)$ and $\kappa(A)$
negative_passages.docid: 76863
negative_passages.text: How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?
negative_passages.docid: 90343
negative_passages.text: Proving the inequality $|e^z-1|\leq e^{|z|}-1$
negative_passages.docid: 137983
negative_passages.text: Normalisation of $k[x,y]/(y^2-x^2(x-1))$
negative_passages.docid: 155884
negative_passages.text: What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?
negative_passages.docid: 138959
negative_passages.text: On the equality case of the Hölder and Minkowski inequalities
negative_passages.docid: 114184
negative_passages.text: Show: $\sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f>k\right\})$
negative_passages.docid: 56958
negative_passages.text: bounding a $C^1$ function that upholds $f'(t)\leq 1-f(t)^2$
negative_passages.docid: 60520
negative_passages.text: Upper bound for $|f(0)|$ based on its behavior within the unit disk.
negative_passages.docid: 121264
negative_passages.text: Prove that rank $(g \circ f)\le \min[\operatorname{rank}(f),\operatorname{rank}(g)]$; $g$ and $f$ are linear mappings
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$
negative_passages.docid: 41477
negative_passages.text: Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
negative_passages.docid: 82371
negative_passages.text: $f$ is an entire function with Im $f\geq 0$
negative_passages.docid: 40930
negative_passages.text: Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$
negative_passages.docid: 97605
negative_passages.text: Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal