Questions tagged [field-theory]

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 topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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3
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1answer
34 views

Proving that multiplicative groups of two fields F* are isomorphic

Considering the fields that are described by: $$\mathbb{F}_i = \mathbb{Z}_2[x] /\langle\mkern 1.5mu p_i(x)\mkern1.5mu\rangle, \enspace i=1,2 $$ where $p_1(x)=x^3 + x + 1$ and $p_2(x)=x^3+x^2+1$ I ...
3
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0answers
23 views

Is my line of reasoning correct for considering this fixed field as a simple extension over $GF(7)$?

Suppose we have the field of rational functions $GF(7)(t)$ in the indeterminate $t$ over $GF(7)$. Define $σ,τ ∈ Aut(E)$ by $σ(t) = 2t$ and $τ(t) = 1 /t$. Set $G = <σ,τ>$ and $F = Fix(G)$. There ...
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1answer
18 views

Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable?

Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable ? I have a feeling that it has something to do with the idea that if $E:F$ is separable hence ...
1
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1answer
29 views

A quicker way to decide if this extension is Galois.

Given $\alpha=\sqrt{1+\sqrt{2}}$, a min. poly for this element over $\Bbb Q$ is $x^4-2x^2-1$, as it's monic irreducible over $\Bbb Q$ and has $\alpha $ as a root. The roots of this min. polynomial ...
0
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1answer
46 views

What theorem is being invoked in this solution to deciding if this polynomial is solvable by radicals?

I was reading these solutions online http://campus.lakeforest.edu/trevino/Spring2019/Math331/Homework7Solutions.pdf for practice for an upcoming exam I have . But in question $2$ part $(a)$ the author ...
1
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0answers
41 views

Is the largest order of an element of a group always the size of the group? [duplicate]

Similar to how one can find a primitive root element of $\mathbb{Z}/p\mathbb{Z}$, I was wondering if, for any group of size $N$ that can be written as $(k\backslash\{0\},\cdot)$ where $k$ is a field, ...
1
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1answer
21 views

Does the maximum number of roots in a field directly imply the maximum number of solutions in a group

From Proposition 2.5 from https://wstein.org/edu/2007/spring/ent/ent-html/node28.html#prop:dsols, the maximum number of roots $\alpha\in k$ of $x^n-1$ in a field $k$ is $n$. That is, there are at most ...
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0answers
28 views

Let $F$ be a field with q elements and $A=F[x_1,..,x_n]$ what is the number of maximal ideals in: [on hold]

The number of maximal ideals in $A$ (I think it is $q$ because the maximal ideal is $x-a_i $, $a_i\in F$ and there are $q$ elements in $F$) Number of maximal ideals $I$ in $A$ that hold $A/I=F$ ...
4
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0answers
50 views

Is there a way to find the subfields of a Galois extension without knowing the subgroup structure of the corresponding Galois group?

Say we have the polynomial $x^4-2$, the splitting field of this over $\Bbb Q$ is $\Bbb Q(\alpha, i)$, $\alpha=\sqrt[4]{2}$, and its Galois group is isomorphic to $D_8$. Now I know a way to find the ...
2
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2answers
37 views

Can an abelian group be a real vector space in more than one way?

We know that any $\mathbb{Z}$ module structure on an abelian group is unique, and furthermore the same is true for $\mathbb{Q}$. Any complex vector space structure is not unique, we can just compose ...
2
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0answers
39 views

Integrating a scalar function on a manifold

So I have the following action in Minkowski spacetime $(M, \eta)$: $ S[\phi] = \int \eta^{\alpha \beta}(\partial_{\alpha} \phi)(\partial_{\beta} \phi)\sqrt{-\eta}d^2x $ Now, I have the following two ...
2
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0answers
19 views

Definition of local and global fields

Reading Neukirch's Algebraic Number Theory I came across the concepts of global and local fields—the first being defined as "a finite extension of either $\mathbb{Q}$ or of $\mathbb{F}_p(t)$ for a ...
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0answers
30 views

separable algebraic closure

Here is the "separable algebraic closure" mentioned. I have heard separable closure and algebraic closure. And I know sometimes separable closures are smaller than algebraiclosures. Does the author ...
3
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1answer
26 views

When notion of polynomial and polynomial function coincide.

Before I never thought about the difference between a polynomial function and polynomial over a certain field $K$. Given examples such as when $K = R$ or $K = C$, we see that sometimes the notion of ...
1
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2answers
19 views

Proving that $\gcd(f(x),g(x))=1$ at a ring and sub-ring [duplicate]

$F$ is a field and $K$ is a sub-field. $K\subseteq F$. $f(x),g(x)\in K[x]$. How to prove that: If in $K[x]$, $\gcd(f(x),g(x))=1$ also at $F[x],\; \gcd(f(x),g(x))=1$? Can you give me a hint? ...
1
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1answer
74 views

Adjoining elements in subextension

Suppose $L/\mathbb{Q}$ is an extension of degree 10 and $\mathbb{Q} \subseteq E_1 \subseteq L$ with $[E_1: \mathbb{Q}]$ = 2 and $\mathbb{Q} \subseteq E_2 \subseteq L$ with $[E_2: \mathbb{Q}]$ = 5. ...
1
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1answer
44 views

My minimal polynomial over this field of rational functions has two indeterminates, please help me understand my error

I was looking over an old exam paper and I came across a question which confused me, it says: Let $E = GF(7)(t)$ be the field of rational functions in the indeterminate $t$ over $GF(7)$. Define $σ,τ ∈ ...
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0answers
51 views

Continuity of the “logarithm” $u \mapsto \frac{du}{udx}$ on function fields

Let $k$ a field and $F$ a finite extension of $k(x)$. Let the rational 1-forms $$Fdx = \{ f dx, f \in F\}=\{ f dg, f,g \in F\} $$ (obeying to the rules of $F$-modules, of $k$-linearity and $d1=0$, $d(...
2
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1answer
45 views

Conjecture about extending a field

My conjecture is motivated by the desire, given a field $F$, to find an extension field by "adjoining" elements. Let's say $\forall x\in F[x*x+1\neq0]$ and we want to "adjoin" an element $α$ ...
2
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0answers
48 views

Notation on Hungerford Algebra Field Theory

I am working through some problems in Field Theory on Hungerford's Algebra. However I have been struggling to understand the notation in some exercises: 1.14 In the field $k(x)$, let $u = x^3/(x+1)...
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0answers
40 views

Algebraic closure of $\Bbb{F}_p(u,v)$

Let $k=\Bbb{F}_p(u,v)$ or any field of the same kind, let $k_s$ be the largest separable algebraic extension of $k$. Claim $\overline{k} =(k_s)^{1/p^\infty}= \bigcup_n (k_s)^{1/p^n}$ For $a \in \...
8
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2answers
122 views

Minimal polynomial of $\sqrt{2+\sqrt[3]{3}}$ over $\mathbb{Q}$

About 2 weeks ago, I tried to solve the following problem. Find the minimal polynomial of $\alpha=\sqrt{2+\sqrt[3]{3}}$ over $\mathbb{Q}$. My attempt First, I tried to find the polynomial with ...
2
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1answer
59 views

Computing $[\mathbb{F}_p(x):\mathbb{F}_p(x)^p]$

If $F=\mathbb{F}_p(x)$ is the rational function field in one variable over $\mathbb{F}_p$, find $[F:F^p]$.* Here's where I am at with this question: Using the multinomial theorem (and some tears), I ...
1
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0answers
31 views

the number of the intermediate fields of a field extension

Let $p$ be an odd prime, $\zeta$ a primitive $p^2$th root of unity and $\alpha = \sqrt[p]{p} \zeta$. Then, (1) Calculate $[\mathbb{Q}(\zeta, \alpha): \mathbb{Q}] $ and $[\mathbb{Q}(\alpha): \...
1
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1answer
37 views

Galois group acts transitively on the roots

Let $F$ be a field and $p(x)$ be an irreducible polynomial in $F[x]$. Let $K$ be a splitting field of $f(x)$. Let $R:=\{\alpha_1,\alpha_2, \alpha_3,\dots,\alpha_k\}$ be the set of all roots of $p(x)$ ...
1
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0answers
40 views

Prime Factors of a Primitive Element

Let $p \in \mathbb{Z}$ be a prime, and let $f(x) = px^n + \dots$ be an irreducible degree $n$ polynomial over $\mathbb{Z}$ with leading coefficient equal to $p$. Suppose that $f(x)$ has no repeated ...
0
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1answer
39 views

Is the subextension of a cyclotomic extension Galois?

If $\alpha$ is a root of the polynomial $f(X)=X^p-a$ over field $F$ of characteristic not $p$, where $p$ is a prime and $a$ is not a $p$-th power. Assume that $F(\alpha) \subset F(\zeta_p)$, do we ...
2
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1answer
17 views

Containment of $\mathbb Q(a)$

Let $a$ be a complex number which is algebraic over $\mathbb Q$. Let $r$ be rational number. Can we say $\mathbb Q(a)$ is contained in $\mathbb Q(a^r)$. My opinion: Since $\mathbb Q(a^r)$ is a field ...
3
votes
1answer
52 views

Show that if Y is transcendental over a field K, and F is a field with $K\subset F\subset K(Y)$ and $K\neq F$ then Y is algebraic on F

I have the following problem: Show that if Y is transcendental over a field K, and F is a field with $K\subset F\subset K(Y)$ and $K\neq F$ then Y is algebraic on F I know there are infinite ...
3
votes
1answer
57 views

Is there a finite field extension with „less“ intermediate fields than Aut-Subgroups?

When looking for a counterexample to the „lattice isomorphism“ part of the fundamental theorem of Galois theory, I usually look for non-Galois extensions $K\colon F$ e.g. by choosing $F$ to not ...
1
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1answer
28 views

$K[X]=K(X)$ with X set of algebraic elements

Let $F$ an extension field of $K$ and $X$ a subset of algebraic elements of $F$ over $K$, $K(X)$ the intersections of all fields containing $K$ and $X$, and $K[X]$ the intersections of all rings ...
1
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0answers
42 views

The splitting field of $x^p-1 \in \mathbb{Q}[x] $ where p is odd prime, contains a unique subfield of index 2

I took particular case when p=3 then I find that $\mathbb{Q}$ is subfield of index 2 in this particular case. I also find that splitting field of given polynomial is $\mathbb{Q(a) }$ where a is pth ...
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0answers
49 views

prove that $[F(\alpha):F]=p$ where $\alpha$ is a root of $x^p-a$ [duplicate]

This is the problem 1.1 from the book, A Gentle Course in Local Class Field Theory. Let $p$ be a prime, let $F$ be a field of characteristic $\neq p$, and let $a \in F^{\times} \backslash F^{\times p}...
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0answers
13 views

Minimal polynomial over a field extension

I came across this question while doing some exercises at the end of the chapter. I would like someone to comment on my solution (on its correctness, completeness and approach): Let $\alpha\in E$, ...
2
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1answer
70 views

Number fields with only trivial field automorphism

Fields with trivial automorphism group have been addressed in a question on Mathoverflow (see this). However, I didn't find sufficient information of number fields with such properties. Q. Let $K$ ...
2
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0answers
18 views

Dedekind zeta function of a real quadratic field

When looking at an imaginary quadratic field, its zeta function is not too difficult to explicitly define (ex: $ \zeta_{\mathbb{Q}(i)}(s) = \frac{1}{4} \sum_{(m,n) \in \mathbb{Z}^2\(0,0)} \frac{1}{(m^...
0
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1answer
42 views

$p$-adic integers and valuation

Let $p$ be a prime, $x \in \mathbb{Z}_p$. I say that if $x \in \mathbb{Z}_p$ then $\text{val}_p(x) \ge 0$ as follows: Let $x = a_0 + a_1 p + a_2 p^2 + \dots$. Then, $\text{val}_p(x) \ge \text{min}(\...
0
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2answers
35 views

Amann/Escher, Analysis I, Exercise I.11.8: field automorphisms of $\mathbb{C}$ which leave the elements of $\mathbb{R}$ fixed

I'm doing Exercise I.11.8 from textbook Analysis I by Amann/Escher. Show that the identity function and $z \mapsto \overline{z}$ are the only field automorphisms of $\mathbb{C}$ which leave the ...
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4answers
42 views

Some confusion about Gaussian ring [closed]

Is $\mathbb{Z} [i] $ is field ? yes/No yes, I thinks it will field because it is integral domain Is its True ?
0
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1answer
24 views

How is $E = S(x)$, where $E = k(x), k$ a field, and S= k(I) of all rationals functions of $I = I(x) = \frac{(x^2 -x+1)^3}{x^2(x-1)^2}$?

In Section G, Part II of Emil Artin's Galois Theory, in the first example.of the section, the author says : Here, $k$ is a field, $E=k(x)$ is the space of all rational functions of variable $x$. We ...
1
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2answers
51 views

Which one is Field

Why $\frac{R[x]}{<x^2-1>}$ is not a field, but if $Q[x]$ is there instead of $R[x]$ it is. How to check $<x^2-1>$ is maximal ideal or not in the easiest way possible?
0
votes
1answer
17 views

Theta function of an ideal

While attempting to find the functional equation for the dedekind zeta function, I encountered this function: $\theta(x_1,x_2,...,x_{r_1+r_2})=\sum_{\alpha \in I_c} e^{-\pi \sum_{k=1}^{r_1+r_2}x_k \...
2
votes
2answers
88 views

Structure of $F=\mathbb Q(\sqrt {2i})$?

Let $F=\mathbb Q(\sqrt {2i})$. My guess would be a field with elements of the form $a+b\sqrt 2i$ with $a,b \in \mathbb Q$. But the last option suggests that it is a vector space. Do not need hints ...
-1
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1answer
42 views

Using the field axioms of real numbers, prove that $y=\frac{1}{x}$ if $xy = 1$ and $x ≠ 0$. [closed]

My attempt: $xy = 1$ and $x ≠ 0 $ Multiplying by $\frac{1}{x}$ $xy \cdot \frac{1}{x} = 1 \cdot \frac{1}{x}$ $y = \frac{1}{x}$ I am not sure if i used the field axioms correctly.
1
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2answers
56 views

Show that $\mathbb{Q}[\sqrt{7}]$ is a field of fractions of $\mathbb{Z}[\sqrt{7}]$ [duplicate]

I've proved that $\mathbb{Q}[\sqrt{7}]$ is a field, but now I have to prove that $\mathbb{Q}[\sqrt{7}]$ is the smallest field containing $\mathbb{Z}[\sqrt{7}]$. What is the best approach to solve ...
1
vote
0answers
42 views

Intuition on embeddings, regulator, discriminant of number fields.

For a very long time I have wanted to derive the class number formula, and recently I was able to wrap my head around the formula in the case of an imaginary quadratic field. Now I want to see if I ...
1
vote
1answer
51 views

Name for operators that preserve structure?

Today in my analytic Calculus course, we discussed the Algebraic and Order limit theorems of sequences: they provide operations like $+$ on both sequences $\{a_n\},\{b_n\}$ and the values they ...
1
vote
2answers
68 views

For $\alpha = \sqrt[3]{2} + i$, find the minimal polynomial over $\mathbb{Q}$ and over $\mathbb{Q}[\sqrt[3]{2}]$

I am trying to find a irreducible polynomial over $\mathbb{Q}[x]$ and over $\mathbb{Q}[\sqrt[3]{2}][x]$ that have $\alpha = \sqrt[3]{2} + i$ as zero. I realized that $p(x) = x^6 + 3x^4 - 4x^3 + 3x^2 +...
4
votes
1answer
82 views

exercise about some field extensions

Let $E = \mathbb{C}(x, y, z)$, $F = \mathbb{C}(x^2y, y^2z, z^2x)$, $L$ the subfield of $E$ fixed by $S_3$, and $K = F \cap L$. Then, (1) Is $E/F$ Galois? And what is its Galois group $G$? (2) ...
0
votes
1answer
48 views

Order of a finite multiplicative subgroup of a field

Let $G$ be a finite subgroup of the invertible elements of a field $F$. Show that if char$F\neq0$, then $G$ is cyclic of order $n$ with $n$ prime to char$F$. I have solved the first part but have no ...