1
vote
1answer
37 views

If $Q$ is a prime ideal of $R[x]$ then $QF[x]\cap R[x]=Q$

I'm filling the gaps in a proof and I'm stuck in this part: Suppose $R$ is a UFD and $Q$ is a prime ideal of $R[x]$, if $F$ is the quotient field of $R$ and $R\cap Q=\{0\}$, then $QF[x]\cap ...
0
votes
0answers
33 views

A question regarding a lemma in Perrin's Algebraic Geometry.

Algebraic Geometry by Perrin says the following: Let $k$ be an uncountable algebraically closed field and let $K$ be an extension of $k$ whose dimension is at most countable. Then $K=k$. He ...
2
votes
1answer
50 views

Proving that a field $K$ can be generated by algebraically independent elements and an separable element

Let $k$ be a perfect field (either $k$ has characteristic $0$, or characteristic $p > 0$ and every element has a $p$th root), and let $K$ be a finitely generated extension field. I have a question ...
6
votes
5answers
304 views

A finite commutative ring with 1 whose elements satisfy a particular equation

I would be very grateful if you give me a hint on it: Suppose $R$ is a finite commutative ring with identity such that $ x^3 = x $ for all elements $x$ of $R$. Then $R$ is a finite direct product ...
0
votes
1answer
38 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the ...
0
votes
1answer
48 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
0
votes
2answers
30 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
3
votes
1answer
41 views

Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
1
vote
1answer
48 views

Commutative ring can be homomorphically mapped onto field

During my algebra lecture, my lecturer used the fact that any commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that? Thanks
4
votes
3answers
184 views

Question about fields and quotients of polynomial rings

I don't see how to solve the following problem: Let $R$ be a commutative and unitary ring. If there exists a monic polynomial $f(x) \in R[x]$ so that $R[x]/(f(x))$ is a field, show that $R$ is a ...
-1
votes
1answer
32 views

Hilbert's Basis Theorem question

If $F$ is a field and $R = F[t_1, t_2, ... t_k]$ and $Y$ is a set of polynomials in $k$ variables over $F$ then by Hilbert's basis theorem apparently $YR = \sum\limits_{i=1}^m f_i R$ for some ...
2
votes
2answers
102 views

What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$?

What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$? This is a question in Hungerford. I understand what both are, $\mathbb{Z}_p = \mathbb{Z}/(p)$ is a finite field and ...
2
votes
1answer
46 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
49 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
35 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
59 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
86 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
69 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 ...
14
votes
1answer
223 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
30 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
57 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
117 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
384 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
104 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
195 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
65 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 ...
6
votes
1answer
176 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
23 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
73 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
48 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
84 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
255 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
107 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
221 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, ...
1
vote
0answers
81 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
174 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
129 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
540 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
187 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
153 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
101 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
203 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
109 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
257 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
54 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
236 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
169 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 ...