Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

learn more… | top users | synonyms

3
votes
3answers
255 views

What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
1
vote
1answer
48 views

If $k[X]/f = k[X]/g$, does $f = g$?

Let $k$ be a field and $f, g$ be irreducible monic polynomials in $k[X]$. Suppose $k[X]/f \stackrel{\sim}{=} k[X]/g$. Then does $f = g$? If so, how can this be generalized? Otherwise, how should I ...
0
votes
0answers
9 views

Norm in field theory

Let $K/k$ be a finite separable extension, and $\sigma_1,\ldots,\sigma_n$ the embedings. For each $\alpha\in K$, the $\textbf{norm}$ is $$Nr(\alpha)=\sigma_1(\alpha)\cdots\sigma_n(\alpha)$$ Then ...
1
vote
1answer
59 views

Addition in Field.

Find counterexamples to the following statements: In every field $\Bbb F$, if $a\in \Bbb F$, $a+a=0$, then $a=0$; Counterexample: Consider $\Bbb Z_2$. Let $a = 1$, so $a + a = 2 = 0 \mod 2$. ...
1
vote
2answers
45 views

Prove the fractional field of an integral domain is the smallest field containing the integral domain

I have two questions about the fractional field of an integral domain. Given an integral domain $D$: Is there a difference between saying "the fractional field of $D$ is the smallest field ...
1
vote
0answers
37 views

Maximality of the kernel of a quasifield

Setting A (left) quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is an abelian group. (As usual, we denote its identity element by $0$.) Each equation $x\cdot a = b$ with ...
2
votes
2answers
21 views

element that is algebraic over a finite field

Let $p$ be a prime. And let $q = p^{2h}$. Suppose I know that an element $\alpha \in \overline{ \mathbb{F}_q }$, satisfies $\alpha^2 + \alpha + 1 = 0$. Does this mean that $\alpha \in \mathbb{F}_{p^2} ...
3
votes
1answer
43 views

Has anyone defined a limit of a sequence of fields? In particular, what is the limit of finite fields?

I'm curious about $$ \lim_{n \rightarrow \infty} \mathbb{F}_n $$ Is it $\mathbb{Z}$? That seems reasonable if you consider it as a set but of course $\mathbb{Z}$ is not a field so that is confusing. ...
2
votes
0answers
38 views

Brauer groups of curves and base change

Let $X/k$ be a smooth, projective curve over $k$ and let $L/k$ be a finite extension of fields, where $k$ is a finite extension of $\mathbb{Q}_p$, $p \not=2$. Suppose $k(X)$ contains no elements ...
2
votes
1answer
30 views

relation between the characteristic polynomial and the minimal polynomial

Define $l_a : F(a) → F(a) $ by $ l_a(x)=ax$, when $[F(a):F]=n$ . show that the minimal polynomial of $a$ over $F$ is the same as the minimum polynomial of $l_a$ as defined in linear algebra. this ...
0
votes
0answers
20 views

algebraic extension fields & intermediate subrings

here is the problem: let $E|K$ be an extension field,prove that it is an algebraic one if & only if every subring $R$ of $E$ containing $K$, is a field. i know "only if" side of proof. but i ...
1
vote
2answers
34 views

Suppose that $L(\alpha):L:K$ and that $[K(\alpha):K]$ and $[L:K]$ are relatively prime.

Show that the minimal polynomial of $\alpha$ over $L$ has its coefficients in $K$. I tried an approach but I got stuck: We have that the following field extensions: $L(\alpha)/L$ and $L/K$ and we ...
1
vote
1answer
43 views

Extension of an Isomorphism

Suppose $E_1, E_2 \subset E$ are proper subfields. In general, if one has an isomorphism $\sigma:E_1\to E_2$, is it possible to extend it to an isomorphism $\psi:E\to E$ s.t. $\psi|_{E_1} = \sigma$ ...
0
votes
2answers
47 views

Prove all elements of $A$ is algebraic over $C$, if all elements of $A$ are algebraic over $B$ and $B$ are algebraic over $C$

Let there be 3 fields $A$, $B$ and $C$. If all elements of $A$ are algebraic over $B$ and all elements of $B$ are algebraic over $C$, prove that this implies that all elements of $A$ is algebraic ...
1
vote
1answer
22 views

a question about field extensions and tower formula

if $K|F$ is a field extension & $a_1,a_2,...,a_n$ are the elements of $K$ which are algebraic on $F$ , we know that $[F(a_1,a_2,...,a_n):F]=<\Pi_{i=1}^n[F(a_i):F]$,it can be proved by induction ...
0
votes
1answer
35 views

Irreducible polynomial iff the condition is satisfied

I am asked to show that the polynomial $f(x)=x^n+1 \in \mathbb{Q}[x]$ is irreducible if and only if $n=2^k$ for any integer $k\geq 0$. Could you give me some hints what I could do to show this??
0
votes
1answer
9 views

Normal closures of transcendental extensions

If $E$ is a finite algebraic extension of the field $F$, then we can find a normal closure of $E$ over $F$. What can we say if the extension $E$ is not finite?
4
votes
1answer
33 views

Fields extensions over isomorphic fields of different degrees

What are the simplest examples of situations where in a field $F$ there are two subfields $L_1$ and $L_2$ such that extensions $F/L_1$ and $F/L_2$ are finite, degrees are different $$ [F:L_1] \neq ...
0
votes
0answers
47 views

The function field of $V=Z(y^2-x^3)$

Let $k$ be a field and let $V=Z(y^2-x^3).$ Can someone explain to me why $k(V)\cong k(s,t)$ ?? with $t=x+(y^2-x^3),s=y+(y^2-x^3)\in A(V)=k[x,y]/(y^2-x^3).$ Can we generalize it : If $V=Z(f)$ with ...
0
votes
1answer
46 views

Show that it is a field

$K \leq E$ an algebraic extension. I am asked to show that each subring of $E$ that contains $K$ is a field. I have done the following: $K \leq E$ algebraic $\Rightarrow \forall a \in E, \exists ...
8
votes
2answers
103 views

${\rm Hom}_R(M, R/M) =\{0\} \implies R$ is a field.

Let $R$ be a local ring with maximal ideal $M$. Suppose $M$ is finitely generated. Prove that if ${\rm Hom}_R(M, R/M) =\{0\}$, then $R$ is a field. ${\rm Hom}_R(M, R/M)$ stand for the group of ...
4
votes
1answer
74 views

On a Proof that the Splitting Field of a Separable Polynomial is Galois

Prop.: If $f \in F[x]$ is separable, then the splitting field of $f$ over $F$ is a Galois extension of $F$. Proof: By induction over $[E:F]$, where $E$ is the splitting field. By previous results ...
0
votes
2answers
31 views

Field extension-degree

I have the following question... $K\leq E$ a field extension. When we have that $$[E:K]=1$$ do we conclude that $K=E$?? Or must also something else be satisfied so that $K=E$ ??
0
votes
2answers
46 views

Field extension-Why does this hold?

$K\leq E$ a field extension, $a\in E$ is algebraic over $K$. Could you explain me why the following holds?? $$K\leq K(a^2)\leq K(a)$$
3
votes
1answer
32 views

When is $0$ mentioned at anytime when talking about Fields, does this mean we are talking about the number $0$, or is it the additive identity?

When talking about fields, such as the field axioms and the theorems that follow, when $0$ is mentioned at anytime, does this mean we are talking about the number $0$, or is it the additive identity?
0
votes
1answer
51 views

How to check field axioms given addition and multiplication tables

I need help with this question, i want to know the exact method of doing it with explanation. i am not able to get around with the logic of it.
2
votes
2answers
36 views

Write it as an element of this ring?

Since the degree of the irreducible polynomial $x^3+2x+2$ over $\mathbb{Q}[x]$ is odd, it has a real solution , let $a$. I am asked to express $\displaystyle{\frac{1}{1-a}}$ as an element of ...
0
votes
1answer
28 views

proof using field axioms only

I am studying the field axioms for real numbers from the book 'Tom m Apostol Calculus' and i am wondering if the following can be proved using the field axioms only. a=a and b=(b). the proofs that i ...
1
vote
1answer
31 views

Galois group of reducible polynomial

I want to find Gaolois group of $(x^3-x+1)(x^2+1)$ over $ \mathbb Q$. The polynomial of degree third is irreducible and has discriminant $-23$ so it's Galois group is $S_3$. Galois group of the other ...
0
votes
1answer
44 views

Algebraic element

$K \leq L, a \in L$ I am looking at the proof that if $a$ is algebraic over $K$, then $K(a)=K[a]$ : We show that $K[a]$ is a field, then we have that $K \subseteq K[a] \subseteq K(a) \subseteq L$. ...
0
votes
2answers
20 views

Homomorphisms between fields are injective.

How would I prove this? I know that I must show f(a)=f(b) => a = b I also know I must use the definition of homomorphism, ie: $f(a+b)=f(a)+f(b)$ $f(ab)=f(a)f(b)$ $f(1)=1$ I am assuming that a ...
0
votes
1answer
44 views

Proof that a given set is a field

I am solving the following exercise (linear algebra): show that: $\ \mathbb{Q}\lbrack\sqrt{2}\rbrack = \{ a + b\cdot\sqrt{2} \ \vert \ a,b \in \mathbb{Q}\} \subset \mathbb{R} \ $ is a field, with the ...
0
votes
1answer
44 views

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$.

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$. First, $\mathbb Q(z)\subseteq \mathbb Q(z^3,z^5)$ would be trivial, right? Then we ...
3
votes
1answer
51 views

Compute the transcendence degree (transcendence degree and tensor products)

$\DeclareMathOperator{\quot}{Quot}\DeclareMathOperator{\tr}{tr}$ Let $I_1$ and $I_2$ be nontrivial ideals in $\mathbb C[x_1,\ldots,x_k]$ and $\mathbb C[y_1,\ldots,y_m]$, respectively. Define $$ R_1 ...
2
votes
2answers
26 views

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$.

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$. I understand that $\mathbb Q(\sqrt{a})$ is the smallest ...
4
votes
2answers
71 views

For what natural numbers $n$ is $\mathbf Z/n\mathbf Z$ $[x]/(x^3+x+1)$ a field?

I recently saw this question in the exam of a first abstract algebra course in my college. It shouldn't be too difficult, yet I can't seem to get the solution. Any ideas on how to tackle this?
1
vote
2answers
66 views

How to show rational function field of an affine subvariety with dim>0 is not algebraically closed?

I do not know how to show the following statement. If $X\subset A^n$ is an irreducible subvariety, $\dim X>0$, then the rational function field of $X$, $K(X)$ is not algebraic closed. What ...
0
votes
2answers
53 views

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$.

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$. One thing to note is $a^{-1}\ne \large\frac{1}{a}$ (same goes for $b$) in this instance as there could be fields where this isn't ...
5
votes
1answer
87 views

History of the three “impossible” compass-and-straightedge problems

I'm preparing a presentation about constructible numbers and I wanted to know some of the history about it to motivate the topic. I wanted to know if the classical Greek problems (doubling the cube, ...
25
votes
1answer
524 views

Prove that both $x+y$ and $xy$ are rational, under some conditions

As a result of the answer I got for this question - Irrational solutions to some equations in two variables - I was wondering if the next statement is always true: Let $x,y$ be real, irrational ...
3
votes
1answer
38 views

Can a $p$-adic field admit a different valuation?

Let $L/\mathbb{Q}_p$ be a finite extension. Question: Is it possible for $L$ to admit a henselian valuation with residue characteristic $q \not=p$? I would think surely not, but I can't see a ...
1
vote
1answer
32 views

Isomorphisms between $\Bbb R^n$ and fields

Are there real vector spaces with dimension $\geq 3$ that are isomorphic to a field? I case $n=2$ there are the complex numbers and for $n=3$ the quaternions are non-commutative. Thanks in advance.
3
votes
0answers
22 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1.
0
votes
2answers
32 views

Solving quadratic equations in the field $F_5$

Let $y = x^2 + 2x + 2 = 0$. Solve the equation in the field $F_5$. So I used the common $b^2 - 4ac$ formula and got that $x$ is either $-1/2$ or $-3/2$ but I'm not sure if this is in the field...
3
votes
2answers
69 views

In the theorem is it necessary for ring $R$ to be commutative?

According to the statement of theorem that a commutative ring $R$ with prime characteristic $p$ satisfies $$\begin{align} (a+b)^{p^n} = a^{p^n} + b^{p^n} \end{align}$$ $$\begin{align} (a-b)^{p^n} = ...
2
votes
0answers
17 views

Is every field between $F$ and $F(\alpha_1,\cdots,\alpha_n)$ of the form $F(\alpha_j,\cdots,\alpha_k)$

Say I have a field $F$, and an extension field $L = F(\alpha_1,\cdots,\alpha_n)$. Is it true that every $K$ such that $$ F \subset K \subset F $$ (all field extensions), $K = ...
1
vote
1answer
29 views

$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : ring of integers, integral basis and discriminant

In the following document, http://people.math.carleton.ca/~williams/papers/pdf/033.pdf, I found three results about biquadratic fields and their ring of integers. It's the proof of the first theorem ...
0
votes
2answers
30 views

Algebraic Extensions

I have the following question: there is this statement i can't understand: Let $A$ be an integral domain which is integrally closed ( in its field of franctions ) and let $K$ be its fraction field. ...
0
votes
1answer
28 views

If $K$ finite field of order $p^8$ where $p\ne3$ then $\sum_{\alpha \in K}{\alpha^2} = 0$

Let $K$ be finite field of order $p^8$ where $p\ne3$ is a prime. Show that $\sum_{\alpha \in K}{\alpha^2} = 0$.
1
vote
3answers
22 views

Isomorphism of field extension

Let $F$ be a field. I need to prove that if $\sigma$ is an isomorphism of $F(\alpha_1,...,\alpha_n)$ with itself such that $\sigma|_F = id_F$ and $\sigma(\alpha_i)=\alpha_i$ for $i=1,...n$, then ...