query: Field extensions - if $(E : F) = n$ then $(E(x) : F(x)) = n$
positive_passages.docid: 143219
positive_passages.text: If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$
negative_passages.docid: 132423
negative_passages.text: Non trivial valuation over a finite extension $F/\mathbb{Q}_p$ is equivalent to the one induced by $v_p$
negative_passages.docid: 63390
negative_passages.text: Infinite algebraic extensions of $\mathbb{Q}$
negative_passages.docid: 51770
negative_passages.text: Number of elements in a finite field extension for finite fields
negative_passages.docid: 127796
negative_passages.text: Degree of field extension $F(x) / F(x^2 + 1 / x^2)$
negative_passages.docid: 111444
negative_passages.text: Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
negative_passages.docid: 82328
negative_passages.text: Constructing a Galois extension field with Galois group $S_n$
negative_passages.docid: 127002
negative_passages.text: Prove that for a nonzero element $a$ of a field $F$ with $q$ elements $a^{-1} = a^{q-2}$
negative_passages.docid: 83009
negative_passages.text: Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
negative_passages.docid: 115129
negative_passages.text: Constructing an explicit isomorphism between finite extensions of finite fields
negative_passages.docid: 16643
negative_passages.text: A slick proof that a field which is finitely generated as a ring is finite
negative_passages.docid: 23946
negative_passages.text: Prove the definitions of $e$ to be equivalent
negative_passages.docid: 115449
negative_passages.text: Inclusion of Fields whose order is a prime power
negative_passages.docid: 62615
negative_passages.text: If F is a finite field, then $F^F=P_F$.
negative_passages.docid: 120943
negative_passages.text: Galois field extension and number of intermediate fields
negative_passages.docid: 66355
negative_passages.text: A sufficient condition for irreducibility of a polynomial in an extension field.
negative_passages.docid: 64571
negative_passages.text: Prove that $\int_{E}f =\lim \int_{E}f_{n}$