# Tagged Questions

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### If an identity in the language of rings holds for all fields, does it necessarily hold for all commutative rings?

It is weirdly difficult to find new identities for ring theory (other than commutativity) that make it more like field theory. This motivates my: Question. If an identity in the language of rings ...
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### Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
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Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} ... 1answer 47 views ### Characterizing Galois field extensions via tensor product Let$K\subset L$be a finite field extension of degree$n$. Prove that it is Galois if and only if$L\otimes_{K} L\simeq L^{n}$, as$L$-algebras, considering on$L \otimes_{K}L=L^{n}$the left ... 3answers 78 views ### The kernel of homomorphism of a local ring into a field is its maximal ideal? I have a question about the proof of Theorem 3.2. of Algebra by Serge Lang. In the theorem$A$is a subring of a field$K$and$\phi:A \rightarrow L$is a homomorphism of$A$into an algebraically ... 0answers 57 views ### Separability of field extensions I'm trying to figure out if these statements are equivalent for an arbitrary field extension$L/k$, such that$\text{char} (k) = p$and$A$is a reduced commutative$k$-algebra.$1)L/k$is ... 1answer 175 views ### Examples of non-isomorphic fields with isomorphic group of units and additive group structure YACP mentions in a comment that: There are examples of non-isomorphic fields$K$and$L$with$(K,+)\cong (L,+)$and$(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$Can someone provide an ... 0answers 28 views ### Proving integrality of the coefficients “inside the box” Consider the (usual)$ABKL$setting:$A$is an integral domain with field of fractions$K$,$L/K$is an algebraic field extension, and$B$is the integral closure of$A$in$L$(we are not assuming ... 1answer 33 views ### When a function field is a regular extension of the field of coefficients? Let$A$be an integral affine$k$-algebra with field of fractions$K$. I am wondering when the extension$K/k$is regular. In particular, is the following statement correct?$K/k$is regular ... 1answer 104 views ### R ring is noetherian, commutative, unitary and integral domain, is R a field? This is the question: "let R be a commutative unitary ring that is also integral domain and noetherian, prove that R is a field" I'm having some trouble proving this. For R to be noehterian means ... 1answer 343 views ### Is every rigid field perfect? A field is rigid iff its automorphism group is trivial. A field$F$is perfect iff all irreducibles in$F[x]$are separable. Is every rigid field perfect? 2answers 99 views ### What is the field of definition of an invariant ideal? Let$K/k$be a finitely generated field extension, such that$k=K^G$for some (possibly infinite) set$G$of automorphisms of$K$. Now, consider the extension of polynomial rings $$... 2answers 178 views ### Spec of tensor product of fields Suppose K/k is a finite separable extension of degree n. How to show that there exists a finite separable extension k'/k such that \operatorname{Spec}(K \otimes_k k') consists of n ... 1answer 62 views ### Finiteness of a field that is a homomorphic image of a polynomial ring Let S=\mathbb F_q[x] be the polynomial ring over the finite field \mathbb F_q. If I=\langle p(x)\rangle is a maximal ideal of S (p(x) is irreducible), then the field S/I is also a finite ... 1answer 164 views ### When is a field a nontrivial field of fractions? If we take any integral domain, then we can define a field of fractions by taking equivalence classes of ordered pairs of elements, the same way that the rational numbers are constructed from the ... 1answer 20 views ### extending an integral domain by an integral element Let A be an integrally closed integral domain, let K be its field of fractions and \bar{K} the algebraic closure of K. Let t \in \bar{K} be integral over A. By a known theorem, the minimal ... 1answer 72 views ### proving closure of a set of automorphisms in the Krull topology in the context of integral ring extensions Let A be an integrally closed integral domain with field of fractions K and let p \in Spec(A). Let L/K be a Galois extension with group G and L'/K be a finite Galois subextension. Let B ... 1answer 42 views ### conjugate prime ideals of integral extensions and relevance of the characteristic of the ground field This question refers to the proof of theorem 9.3, p. 66 in Matsumura's Commutative Ring Theory: "if A is an integrally closed domain, K its field of fractions and L/K a normal field extension, ... 2answers 63 views ### Ideal norm in a quadratic field Let K=\mathbb{Q}[\sqrt{d}] be a quadratic field with discriminant d_K, let \mathfrak{a}=(a,\frac{b-\sqrt{d_K}}{2}) be an ideal. Does the norm N(\mathfrak{a})=a? How to prove it? 2answers 249 views ### Extension of residue fields and algebraic independence Let A be a Noetherian integral domain, B a ring extension of A that is an integral domain, P \in \operatorname{Spec} B, \, p = P \cap A. Denote by \kappa(p),\ \kappa(P) the residue fields of ... 1answer 98 views ### Valuation ring in an algebraically closed field Let k be an algebraically closed field and (R, \mathfrak{m}) a valuation ring in k i.e. the field of fractions of R is k. Then Mumford (Red Book, page 127) claimed that the residue field ... 1answer 178 views ### A subset of a field that is a subfield It can be verified that the following assertion is true: a subset S of a field F is a subfield if S contains the additive and multiplicative identities 0 and 1, if S is closed under addition, ... 0answers 70 views ### Hilbert’s zeros theorem, an application. (The algebraic variation) Theorem: (Hilbert) If k is a field, A is a finitely generated k-algebra, and M is a maximal ideal in A, then the factor A/M is a finite extension of k. In particular if k is ... 1answer 133 views ### Finitely generated integral domain and finitely generated k-algebra. Let k be a field and R a finitely generated domain over k. Then there are y_1, \ldots, y_n such that$$R=k[y_1, \ldots, y_n]/P,$$where$P$is some prime ideal of$k[y_1, \ldots, y_n]$. My ... 3answers 116 views ### Lifting isomorphisms of fields to automorphisms of polynomial rings Let$L$be a field and$\alpha, \beta$algebraic over$L$such that$L(\alpha)\cong L(\beta)$. If$q(t)$and$p(t)$are the minimum polynomials of$\alpha$and$\beta$, respectively, does it follow ... 2answers 434 views ### Tensor products of fields Let$K/F$be a field extension. I am interested in the situation where there exists a field extension$L/F$such that the ring$L \otimes_FK$is not a field. If there exists$z\in K \setminus F$... 0answers 182 views ### Purely transcendental field extensions and free composite Let$k$be a subfield of two fields$E$and$F$. If a field extension$L/E$is purely transcendental, is it true that the field extension$F.L/F.E$obtained by taking the free composite of field ... 1answer 143 views ### Examples of extensions of a perfect field which are not separably generated Let$K$be an extension field of a field$k$. We say$K$is separably generated over$k$if$K$has a transcendence basis$S$over$k$such that$K$is separably algebraic over$k(S)$. Let$k$be a ... 1answer 98 views ### Question about extensions of homomorphisms I have difficulty understanding the proof of Theorem 3.2 in Lang's Algebra Chapter VII. Let$A$be a subring of a field$K$and let$x\in K, x\neq 0$. Let$\phi:A \rightarrow L$be a ... 1answer 64 views ### Extension of some properties of$\mathbb{R}$to other fields and subrings. We know that the only non-zero ring homomorphism from$\mathbb{R}$to$\mathbb{R}$is identity. From this some questions came in to my mind as follow: Question$1$: Can we characterize all fields ... 1answer 54 views ### A question regarding solutions of polynomials in a field Let$F$be a field and$\langle a_1,...,a_n \rangle \subset F$. Then given a non-zero polynomial$f \in F[X_1,...,X_n]$is it true that if$f(a_1,...,a_n)=0$then$(X_i - a_i)$divides$f$for some ... 1answer 180 views ### Transcendence degree for a$k$-algebra which is an integral domain Let$R$be an integral domain over a field$k$. Is it true, that$\deg.\mathrm{tr}_k \ \mathrm{Frac}(R)$is the greatest number of elements of$R$algebraically independent over$k$? 1answer 104 views ### Maximal ideals of$k[x_1,\cdots,x_n]$and degrees of field extensions For the past couple of days i had been working on an interesting homework problem, my interpretation of which is as follows: Let$m$be a maximal ideal of$k[x_1,\cdots,x_n]$. Let$m_i = m \cap ...
Suppose $L,K$ are fields. Is is true that if $L$ a finitely generated $K$-algebra then $L/K$ is a finite field extension? Wikipedia seems to think so. But if it is true surely it's difficult to ...