2
votes
1answer
37 views

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 ...
2
votes
2answers
41 views

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 ...
0
votes
1answer
19 views

Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

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} ...
2
votes
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 ...
1
vote
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 ...
1
vote
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 ...
11
votes
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
16
votes
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?
4
votes
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 $$ ...
6
votes
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 $n$ ...
2
votes
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 ...
5
votes
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 ...
0
votes
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 ...
1
vote
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 $B$ ...
0
votes
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, ...
2
votes
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?
6
votes
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 ...
5
votes
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 ...
4
votes
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, ...
2
votes
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 ...
1
vote
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 ...
5
votes
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 ...
1
vote
2answers
433 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 $z\in K \setminus F$ ...
1
vote
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 ...
5
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
3
votes
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 $k$?
1
vote
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 ...
1
vote
1answer
236 views

Finitely Generated Algebra and Finite Extension

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 ...
0
votes
1answer
53 views

What is known about moduls $M = F^n$ over a ring $R$ where $F = R/I$ is a field

If $R$ is a ring and $I$ is an ideal of $R$, then $F = R/I$ is a homomorphic image of $R$, i.e. there is a homomorphism $f: R \rightarrow F$. If you let $M = F^n$, and define $(\cdot): (R,M) ...
2
votes
0answers
225 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
3
votes
1answer
154 views

Commutative Algebra - Polynomial Rings

Let $Z$ be the ring of integers, $p$ a prime and $F_p = Z/pZ$ the field with $p$ elements. Let $x$ be an indeterminate. Set $R_1 = F_p[x]/(x^2-2)$, $R_2 = F_p[x]/(x^2-3)$. Determine whether the rings ...
2
votes
1answer
61 views

Extending $\phi: A \rightarrow \Omega$ to $A[x] \rightarrow \Omega$ where $A$ is integral domain and $x$ transcendental over $A$

Let $A \subseteq B$ be integral domains and let $\phi:A \rightarrow \Omega$ be a homomorphism of $A$ into the infinite algebraically closed field $\Omega$. Let $x \in B$ and suppose that $x$ is ...
7
votes
1answer
237 views

Extension of Homomorphisms (Lang, Atiyah and McDonald)

Let $A$ be a subring of a field $K$, and suppose that $A$ is a local ring with maximal ideal $\mathfrak{m}$. Let $x \in K, \, x \neq 0$. Let $\phi: A \rightarrow L$ be a homomorphism of $A$ into the ...
4
votes
2answers
297 views

A question about a proof of a weak form of Hilbert's Nullstellensatz

I'm trying to prove the following (corollary 5.24 page 67 in Atiyah-Macdonald): Let $k$ be a field and let $B$ be a field that is a finitely generated $k$-algebra, i.e. there is a ring homomorphism ...
3
votes
2answers
359 views

A question about a weak form of Hilbert's Nullstellensatz

Corollary 5.24 on page 67 in Atiyah-Macdonald reads as follows: Let $k$ be a field and $B$ a finitely generated $k$-algebra. If $B$ is a field then it is a finite algebraic extension of $k$. We know ...
0
votes
3answers
196 views

$x$ algebraic over $K$, $v$ a polynomial in $x$ then $v$ algebraic?

In the proof of proposition 5.23 Atiyah-Macdonald on page 66 use that if $x$ is algebraic over $K$ and $v = a_n x^n + \dots + a_1 x + a_0$ then $v$ is algebraic over $K$ (where $K$ is the field of ...
6
votes
3answers
840 views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
20
votes
2answers
526 views

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
2
votes
1answer
68 views

Under some conditions, $K$ is algebraically closed in $K(x, y)$

Let $K$ be a field and $L = K(x, y)$, where $x$ is transcendental over $K$ and $y$ is such that $f(x, y) = 0$, for $f \in K[X, Y]$ irreducible. I have to prove that if $f$ is also irreducible over ...
2
votes
2answers
300 views

Isomorphism of tensor product+field extension

Let $k$ be a field, $f(x)\in k[x]$ be an irreducible polynomial over $k$, and $\alpha$ be a root of $f$. If $L$ is a field extension to $k$, what does $k(\alpha)\otimes_k L$ isomorphic to? I'm ...
7
votes
3answers
166 views

Every element is radical in a field extension.

Let $L/K$ be an algebraic field extension. Suppose for each $x\in L$, there exists an integer $n>0$ such that $x^n\in K$, where $n$ may depend on $x$. If the characteristic of $K$ is zero, does it ...
2
votes
1answer
149 views

A question with an odd hypothesis.

Let $S$ be a discrete valuation ring and $R\subset S$ be a proper subring (also a DVR). Assuming that $M$ and $N$ are the respective maximal ideals of $R$ and $S$ and that $N\cap R = M$, then the ...
2
votes
0answers
83 views

Intersections of finitely generated field extensions are finite?

I was reading the following post at MathOverflow: http://mathoverflow.net/questions/21086/when-are-intersections-of-finitely-generated-field-extensions-finitely-generated/21093 I can't comment there, ...
5
votes
2answers
507 views

A slick proof that a field which is finitely generated as a ring is finite

It is a known fact that if $k$ is a field that is finitely generated as a ring, which is the same as having a surjective ring homomorphism $f:\mathbb{Z}[x_1,\dots,x_n]\to k$ for some $n\in ...