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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2
votes
1answer
38 views

Suppose $F$ is a field and $F \subset \mathbb C$.Then does $F^3$ contain $(1,0,0),(0,1,0), (0,0,1)$.

[Note:I have yet not studied field theory,I am just using it in vector spaces.So my question may look silly.] Suppose $F$ is a field and $F \subset \mathbb C$.Then does $F^3$ contain $(1,0,0),(0,1,0), ...
1
vote
2answers
302 views

If $\alpha$ separable over $F$ then $F(\alpha )/F$ is a separable extension.

Let $K/F$ be a field extension and $\alpha \in K$ is algebraic over the field $F.$ Now suppose $\alpha$ is separable over $F.$ Then how can I show that $F(\alpha)/F$ is a separable extension, i.e., an ...
0
votes
1answer
31 views

Does an algebraic closure of $F_p$ contain an element of infinite (multiplicative) order?

I am trying to find (as many as possible) elements in the algebraic closure of a positive characteristic field, being roots of irreducible polynomial inside some splitting field which are not roots of ...
0
votes
2answers
40 views

Splitting Fields over arbitrary fields

I am currently learning field theory by myself (following Advanced Modern Algebra by Joseph Rotman at Chapter 3). We suppose $\mathbb{F}$ is some arbitrary field and $P\in\mathbb{F}[x]$, the ...
2
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2answers
51 views

Why $s(1-s)$ numbers which are squares in a field are written $\frac{u^2}{1+u^2}$?

Trying to exercise in Math again after years of other activities, I need a little help on this : Let $F$ be some arbitrary field with characteristic > 2. Let $S \subseteq F$ be the set of numbers s ...
1
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1answer
37 views

Rational functions over $\mathbb{C}$ is algebraically closed

Studying for a course in fields, and came across the question: is the field $\mathbb{C}(x)$ of rational functions over $\mathbb{C}$ algebraically closed? At the moment I think it is, as I can't seem ...
2
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0answers
33 views

Finding polynomials that can be iterated to generate the field.

For a prime p and a field $\mathbb{Z}_p$, is there support to find polynomials in $\mathbb{Z}_p[x]$ that can be iterated to generate all of $\mathbb{Z}_p$? As an example that such exist, let p = ...
1
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3answers
38 views

If $R$ is a ring, $K$ a field (and subring of $R$), and $I$ a proper ideal of $R$, $R/I$ contains a field isomorphic to $K$

Let $R$ a ring, $K$ subring of $R$ and $I$ a proper ideal of $R$. Now suppose $K$ is a field. I need to prove that $R/I$ contains some field isomorphic to $K$. My idea is to take $K/I$ as that ...
2
votes
1answer
32 views

Place at infinity in function fields

I've just started reading about function fields. If I understood it correctly, for a function field in one variable $F|K$ there is a correspondence between places (i.e., maximal ideal of a valuation ...
1
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0answers
36 views

Algebraic Extension Question [on hold]

If $K$ is an algebraic extension of $\mathbb{Q}(\sqrt{2})$, prove $K$ is also an algebraic extension of $\mathbb{Q}$. Can someone kindly provide some ideas to approach this question? Thanks a lot.
8
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3answers
3k views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
6
votes
2answers
91 views

There does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F) $ for any field $\mathbb F.$

For a positive integer $n$, I have to show that there does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F) $ for any field $\mathbb F.$ If it ...
7
votes
5answers
220 views

Show that $\sqrt[3]{1+\sqrt{3}}$ isn't an element of the field $\mathbb{Q}(\sqrt{3} ,\sqrt[3]{2})$

Setting $\alpha = \sqrt[3]{2}$ and $a+b\alpha+c\alpha^2=\sqrt[3]{1+\sqrt{3}}$ for $a,b,c$ in $\mathbb{Q}(\sqrt{3})$ (The minimal polynomial for $\sqrt[3]{2}$ in $\mathbb{Q}(\sqrt{3})$ is $x^3-2=0$ ...
1
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0answers
44 views

$S$ is a commutative integral domain and a finitely generated $R$-Module where $R$ is a subring of $S$. $R$ is a field iff $S$ is a field

Assume $S$ is a commutative integral domain, and $R \subseteq S$ is a subring. Assume $S$ is finitely generated as an $R$-module, i.e., there exist elements $s_1, \ldots, s_n \in S$ such that $S = ...
0
votes
1answer
18 views

In ZFC there exists a perfect field of given positive characteristic and given infinite cardinality

Fix a prime number $p$. In ZFC does there exist a perfect field of characteristic $p$ of any infinite cardinality? I know some constructions of fields of characteristic $0$ of arbitrary cardinality ...
2
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0answers
31 views

Why $X$ is a primitive r-th root of unity in lemma 4.7 Prime is in P? [duplicate]

Let $Q_r(X)$ be the $r^{th}$ cyclotomic polynomial over $F_p$. Polynomial $Q_r(X)$ divides $X^r-1$ and factors into irreducible factors of degree $o_r(p)$. Let $h(X)$ be one such irreducible factor. ...
0
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0answers
20 views

Question about finite extensions and splitting fields

Apparently any finite extension of a field with characteristic 0 is of the form $F(a)$ for some algebraic $a$. But wouldn't this mean that any two zeros of a polynomial (e. g. over $\mathbb{Q}$) would ...
1
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1answer
56 views

Show that $\mathbb{Q}(\sin\theta)$ is a field

QUESTION: Show that $F_{\theta}=\mathbb{Q}(\sin\theta); \theta\in\mathbb{R}$ is a field. Moreover, Show that $E_{\theta}=\mathbb{Q}(\sin\frac{\theta}{3}); \theta\in\mathbb{R}$ is a field extension ...
1
vote
1answer
37 views

Is there an algebraic closed field which contains the complex field $\mathbb{C}$ strictly?

So we know that $\mathbb{C}$ itself is algebraically closed. But I was thinking if there maybe is an algebraically closed field which contains $\mathbb{C}$ strictly, so that it is not equal to $\...
2
votes
1answer
53 views

Galois Group of $\mathbb{Q}(\sqrt{\sqrt{3}-1})$

I am reviewing Galois Theory questions for an upcoming prelim, and I came across this question on a past prelim "Find the Galois Group $\operatorname{Gal}(\mathbb{Q}(\sqrt{\sqrt{3}-1})$" The minimal ...
0
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0answers
19 views

Bases consisting of norms and traces

$\newcommand{\gal}{\text{Gal}}$ I am trying to give a detailed proof of the following facts: If $L$ is an intermediate field of a Galois extension $K/F$, then it is generated by traces or norms of $...
0
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0answers
19 views

Residue field and value group of an algebraically closed valued field [closed]

Why is the residue field of an algebraically closed valued field algebraically closed and the value group is divisible?
0
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1answer
43 views

Proof Explanation *Field and Galois Theory*

The following is with regards to Lemma 3.18 from Field and Galois Theory by Patrick Morandi. Let $\sigma : F \to F'$ be a field isomorphism, $K$ be a field extension of F, and $K'$ be a field ...
2
votes
2answers
46 views

Elements of $E^{\times},\cdot$ of the quotient ring $E:= \frac{\mathbb{Z}_3[X]}{\langle x^2 + x + 2\rangle}$

Consider the field $E:= \frac{\mathbb{Z}_3[X]}{\langle x^2 + x + 2\rangle}$. If I'm right the elements of the quotient ring can be found as: $$a_0 + a_1x + \langle x^2 + x + 2\rangle.$$ So we got the ...
1
vote
1answer
43 views

Galois group of order $n!$

Suppose $f\in\mathbb{Q}[x]$ has degree $n$, and let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Suppose the Galois group of $K/\mathbb{Q}$ is $S_n$ with $n\geq 3$. It is not hard to show that ...
0
votes
1answer
27 views

$(M\cap E)(M\cap F)= M$ for linearly disjoint fields $E$ and $F$?

Let $E$ and $F$ be linearly disjoint fields over a base field $K$ (all contained in an algebraic closure $\overline K$). Suppose there is an extension $M/K$ contained in $EF$. Is it true that $(M\cap ...
3
votes
1answer
33 views

Prove that $[\mathbb{Q}(\sqrt{3},i):\mathbb{Q}=4$. Find a number $a$, such that $\mathbb{Q}(a)=\mathbb{Q}(\sqrt{3},i)$

The first part follows from the fact that for any algebraic elements $a_i$ the following holds: $$F(a_1,...,a_n)=(F(a_1,...,a_k))(a_{k+1},...,a_n)$$ where $F$ is the field generated by the $a_i$. ...
1
vote
3answers
26 views

Degree of the splitting field over finite field

Let the K be a splitting field of $f(x) =x^3 + 5x+ 5$ over $Z_3 $ What is the $[K;Z_3]$ ? The solution in my book) Let the $\alpha$ be a solution of the $f(x$) Since $K = Z_3 (\alpha$) ...
0
votes
1answer
28 views

Let $K/F$ be a field extension. If $\alpha \in F(\alpha^m)$, $m > 1$, then $\alpha$ is algebraic in $F$.

Let $K/F$ be a field extension. If $\alpha \in F(\alpha^m)$, $m > 1$, then $\alpha$ is algebraic in $F$. A proof provided in the book is as follows: Proof: Since $\alpha \in F(\alpha^m)$, there ...
0
votes
0answers
48 views

$X^{p^n} - a$ is irreducible if $a \in K$ has no $p$-th root? [duplicate]

Exercise from Lang. I am trying to show that if $char K = p$ and $a \in K$ has no $p^{th}$ root (implying that K is an infinite field) then $X^{p^n} - a$ is irreducible for all positive integer $n.$ ...
1
vote
2answers
197 views

Prove that a polynomial is irreducible or the field contains a $p$th root

An exercise from Lang: Let $k$ be a field. Let $n$ be a nonnegative integer. Consider the polynomial $x^{p^n}-a \in k[x]$. Assume that $\text{char}(k) = p > 0$. Prove that either ...
2
votes
1answer
34 views

Finding elements that are algebraic over a given field

Find all $k \in \mathbb{N}$ such that there exist elements in the field $\mathbb{Q}(\sqrt[\leftroot{-2}\uproot{2}12]{2)}$ that are algebraic of order $k$ over $\mathbb{Q}$. For each such $k$ find an ...
1
vote
3answers
57 views

If every non-zero ring homomorphism $\phi: K\rightarrow S$ is injective, $K$ is field

If every ring homomorphism $\phi: K\rightarrow S$ is injective, $K$ is field. PS: $K$ is a commutative ring with unity, and my definition of homomorphism includes $\phi(1_{K})=1_{S}$. My idea is to ...
-3
votes
1answer
82 views

Number of elements in a subset of $\mathbb F_{27}$ [duplicate]

Let $\mathbb F_{27}$ be the finite field of order $27$. Let $A_{a}=\{1,1+a,1+a+a^2,\dots\}$. Then which of the following are true? The number of $a\in\mathbb F_{27}$ such that $\operatorname{card}(A_{...
0
votes
2answers
34 views

In a finite field there exists an irreducible polynomial of degree at least $n$ for $n \in \mathbb{N}$

This question already has an answer here, but I'm looking for a solution that doesn't use field extensions. It's relatively easy to find a polynomial without zeros for every $n$ but as far as I know ...
2
votes
1answer
137 views

A finite field problem

$\mathbb F$ is finite field, $\mathbb A$ is a subset of $\mathbb F$, and $|\mathbb A|>|\mathbb F|^{\frac 34} $. Proof $\forall x \in \mathbb F$, there exist $a,b,c,d,e,f\in\mathbb A$ that makes $x ...
6
votes
0answers
61 views

Normal extension $\Longleftrightarrow$ splitting field extension, intuitive or counterintuitive? [on hold]

$\textbf{Question}$: Would you say that normal extension $\Longleftrightarrow$ splitting field extension is intuitive or counterintuitive? A splitting field extension $L:K$ is exactly large enough to ...
2
votes
1answer
32 views

Determine the irreducible polynomial

$$ \newcommand{\abs}[1]{\left\vert #1 \right\vert} \newcommand\rme{\mathrm e} \newcommand\imu{\mathrm i} \newcommand\diff{\,\mathrm d} \DeclareMathOperator\sgn{sgn} \renewcommand \epsilon \varepsilon \...
3
votes
1answer
47 views

Showing $\mathbb{Q}(\sqrt{5+2\sqrt{6}}) = \mathbb{Q}(\sqrt{2},\sqrt{3})$

I am reviewing Galois theory for my algebra prelim, and I got stuck on the following problem. Prove that $\mathbb{Q}(\sqrt{5+2\sqrt{6}}) = \mathbb{Q}(\sqrt{2},\sqrt{3})$. Deduce that $K$ is a normal ...
3
votes
4answers
120 views

Simplify $ \frac{ \sqrt[3]{16} - 1}{ \sqrt[3]{27} + \sqrt[3]{4} + \sqrt[3]{2}} $

Simplify $$ \frac{ \sqrt[3]{16} - 1}{ \sqrt[3]{27} + \sqrt[3]{4} + \sqrt[3]{2}} $$ Attempt: $$ \frac{ \sqrt[3]{16} - 1}{3 + \sqrt[3]{4} + \sqrt[3]{2}} = \frac{ \sqrt[3]{16} - 1}{ (3 + \sqrt[3]{4}) +...
4
votes
1answer
37 views

Given an arbitrary ordered field (F,<), is there always a sequence of strictly positive elements of F that tends to zero?

Let $(F,<)$ be an ordered field. Is there always a sequence $(a_n)_{n\in\mathbb N}$ of strictly positive elements of $F$ such that $\lim_{n\to\infty}a_n=0$? (To define limits, I'm using the ...
21
votes
1answer
435 views

Galois correspondence and characteristic subgroups

It is well-known that Galois correspondence sends a normal subgroup to a normal extension of a field. Specifically, given a Galois extension $L/K$ and the corresponding Galois group $G$, normal ...
0
votes
2answers
37 views

Why is $k-\{0\}$ not a closed in $k$ in the zariski topology?($k$ algebraically closed)

Why is $k-\{0\}$ not a closed in $k$ in the zariski topology?($k$ algebraically closed) I am trying to prove that if $h:X\to Y$ is a morphism of varieties, then $h(X)$ is not necessarily a subvariety ...
0
votes
2answers
24 views

How do I define a multiplication operator here so that this mapping is isomoprhic? [closed]

Part a) of this is fine, but I'm really stuck on part b) and I have a test on this in an hours time, does anyone have any hints?
3
votes
2answers
57 views

A general question on Galois theory: cyclotomy

Say we that we have a $pth$ root of unity where $p>2$ is prime. Let $E=\Bbb Q(w)$ and let $\alpha=w+w^{-1}$. I want to figure this question out asked on a past exam: 1) Compute | E : $\Bbb Q(α)...
4
votes
4answers
298 views

prove that if $a=b$ then $a+c=b+c$ where $a,b,c\in \mathbb R$

I was trying to prove if $l=m$ and $m=n$ then $l=n$ but when doing this I had to add $-m$ to both sides of both equations.i think it is not appropriate to proceed without proving "if $a=b$ then $a+c=b+...
1
vote
0answers
35 views

When is the Galois group of a quintic not $S_5$ if we use this particular method?

I think that I am misunderstanding something fundamental about the technique used to decide if higher order polynomials are solvable by radicals using Galois theory. If we have a cubic it's not to bad ...
1
vote
3answers
57 views

$\frac{\mathbb{C[x]}}{\langle x-a\rangle}$ is isomorphic to which field?

We know that every polynomial of degree one is irreducible over $\mathbb{C}$ that is $\langle x-a \rangle$ is maximal ideal in $\mathbb{C[x]}$, hence $$\frac{\mathbb{C[x]}}{\langle x-a\rangle}$$ is ...
6
votes
2answers
134 views

Is every $1$-dimensional vector space a field?

We say that every field $F$ "is" a $1$-D vector space over itself. By this we mean that if we consider the elements of $F$ as both vectors and scalars, then we get a vector space by using the addition ...
2
votes
2answers
47 views

If $a$ is a sum of squares then there exists an element whose norm is $-1$.

This is part of a question(2.9.3) from Patrick Morandi's "Field and Galois Theory" I am just plain stuck on this fact: $F(\sqrt{a})/F$ is a field extension where $a\in F-F^2$ and $F$ does not ...