Tagged Questions

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 ...

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0
votes
2answers
62 views

Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a\cdot a^p$ Let $a\cdot a^p = b\cdot ...
-3
votes
2answers
102 views

What are the root of $x^3 - 2$ $\in \mathbb{R}[x]$? [closed]

From the given polynomial it is evident that we have to find a +ve number in $\mathbb{R}$ such that the cube is 2 if it exists. There is one and that is $\sqrt[3]{2}$. How to find the other roots in ...
0
votes
1answer
32 views

Rational function in both $k(X)[Y]$ and $k(Y)[X]$

If I have a rational function in $X$ and $Y$ and it can be written as both a polynomial in $Y$ with coefficients being rational functions in $X$ (that is, an element of $k(X)[Y]$) and as a polynomial ...
1
vote
1answer
39 views

Write Galois group as semidirect-product

Consider the polynomial $ x^7-13 \in \Bbb{Q}(x)$. Find Galois group and write it as a semi-direct product. Edit: So here is what I have done. I found the dimension of the splitting field over ...
0
votes
4answers
63 views

Why has my prof worked out the multiplicative inverse of complex numbers this way?

She explains how to obtain the multiplicative inverse: In what follows, z=a+bi and w=c+di are complex numbers with a,b,c,d∈R. Is there a multiplicative inverse of z? If so, what is it? Note ...
0
votes
1answer
17 views

Is every element in a finite splitting field K over F a root in a polynomial?

Let $ K$ be a finite extension of $F$ and assume $K$ is a splitting field over $F$. Is it given that for any element $ \alpha \in K, \alpha \not\in F $ that there exists a polynomial $ f(x) \in F[x] $ ...
2
votes
2answers
73 views

When can an infinite abelian group be embedded in the multiplicative group of a field?

This question comes from this question by user72870. That question would easily be answered if we know the cyclicity of the group in question, but, as the OP appears to be trying to prove that the ...
3
votes
2answers
88 views

Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...
2
votes
0answers
26 views

Finding degree of an extension

Find the degree of the field extension $\mathbb{Q}[\sqrt[3]{2},\sqrt[3]{3}]$ over $\mathbb{Q}$. My approach: Call the desired degree $n$. Clearly, $3|n$ and $n\leq 9$. So possible values of $n$ are ...
0
votes
1answer
47 views

how do you know if a collection of subsets is a field?

Let $\Omega=${0,1,2,3,...} Let B be the collection of subsets of $\Omega$ such that C $\in$ B if and only if either C or $C^{c}$ is a finite set. Is B a field? Is it a $\sigma$-field? Here is my ...
3
votes
2answers
38 views

example of two non-isomorphic fields which embed inside each other

Can you find an example of non-isomorphic fields which embed inside each other? Most probably we can't but I am looking for extraordinary answer...
2
votes
0answers
21 views

How do I show that both the additive and multiplicative groups of an infinite field are non-cyclic? [duplicate]

I've tried mimicking the proof in case of $\mathbb Q$ to deal with the characteristic zero case, but can't do the characteristic $p$ case. Can someone give a solution to that end?
2
votes
1answer
44 views

Field Theory : What is wrong with this “homomorphism”?

Let $E/F$ be a field extension and $\alpha \in E$ be algebraic over $F$. Let $m(x) \in F[x]$ be irreducible and such that $m(\alpha) \neq 0$. Define $$ \phi : F(\alpha) \to F[x]/(m) $$ by ...
1
vote
1answer
20 views

If a ring has its field of fraction as algebraic number field $K$, would this ring be $O_K$?

Suppose that ring has its field of fraction as algebraic number field $K$. Would this ring then be $O_K$, ring of integers? Also, for $O_K$, would subring of $O_K$ be integrally closed?
35
votes
6answers
3k views

In plain language, what's the significance of a field?

I just started Linear Algebra. Yesterday, I read about the ten properties of fields. As far as I can tell a field is a mathematical system that we can use to do common arithmetic. Is that correct?
1
vote
0answers
30 views

Number fields and algebraic integer

Suppose there is a (algebraic) number field $K$. Algebraic integer is a root of some monic polynomial with coefficients in $\mathbb{Z}$. The elements of $K$ are the root of some monic polynomial ...
0
votes
1answer
34 views

Proof Unicity of Splitting Field

Context : Definition : We say $f(x) \in F[x]$ splits over the field extension $E/F$ if $$ f(x) = c (x-\alpha_1)\cdots(x-\alpha_n) $$ for some $c, \alpha_1, \ldots, \alpha_n \in E$. A splitting ...
2
votes
0answers
36 views

Is this element constructible from this elements?

Let the figure below. According to same notation of the figure verify if it's possible to construct the point $\displaystyle \zeta=e^{\frac{2\pi i}{13}}$ with straight-edge and compass from ...
4
votes
1answer
50 views

Infinite algebraic extension of $\mathbb{Q}$

I have this problem in a exercise list: "Prove that $K=\mathbb{Q}(2^{\frac{1}{2}},2^{\frac{1}{3}}, 2^{\frac{1}{4}}, \ldots)$ is an algebraic extension, but not a finite extension of $\mathbb{Q}$." ...
2
votes
1answer
28 views

$\alpha \in \Omega_{\mathbb Q}^{x^3-2}$(splitting field) is such that $\alpha^5 \in \mathbb Q$ then $\alpha \in \mathbb Q$

Let $f(x)=x^3-2$ and let $E$ be it's splitting field. Prove that if $\alpha \in E$ is such that $\alpha^5 \in \mathbb Q$ then $\alpha \in \mathbb Q$. Let $\zeta_n$ denote a n-root of unity. This is ...
0
votes
3answers
48 views

clarification of algebraic closure and algebraically closed field

Definition of Algebraic closure: An extension K of F is called an algebraic closure of F if a) F $\subset$ K is algebraic b) K is algebraically closed given the above definition I have been trying ...
3
votes
2answers
42 views

$F(x,y)$ over $F$ is not simple

Let $F$ be a field. Let $x,y$ two algebraically independent indeterminates. Show that $F(x,y)/F$ is not a simple extension. Attempt: I tried by contradiction, assuming that $F(t)=F(x,y)$ and writing ...
2
votes
0answers
53 views

When $\mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta)$?

Let $\alpha, \beta$ be algebraic numbers over $\mathbb{Q}$. Which necessary and sufficient conditions are known such that $$ \mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta) \text{ ?} \tag{$\ast$}$$ ...
4
votes
3answers
63 views

$[F(t):F(t^n)]=n$ where $t$ is trascendental

Let $F$ be a field and let $t$ be trascendental over $F$. Prove that $[F(t):F(t^n)]=n$. Obviously $[F(t):F(t^n)]\le n$ since the polynomial $f(x)=x^n-t^n \in F(t^n)$ has $t$ as a root. But I don't ...
2
votes
1answer
67 views

Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients ...
0
votes
0answers
27 views

On the Characterization of $F[\alpha]$ for a Field Extension $E/F$.

Let $E/F$ be a field extension. Reading a proof that $$ \alpha \text{ is algebraic over } F \implies [F[\alpha]:F] < \infty, $$ one might begin by considering an element $f(\alpha) \in F[\alpha]$ ...
2
votes
2answers
75 views

Can the product of only some of the algebraic conjugates be an integer?

Suppose I know that $x_1,\dots, x_n$ are algebraic conjugates and suppose that their product is a rational integer: $$ \prod_{i=1}^{n}x_i\in \mathbb{Z} $$ Is it possible that there exists some other ...
0
votes
2answers
146 views

Give an example of a field where -1=1

The question is to find a counterexample to the following: In every field the element $-1$ is not equal to $1$. My intuition leads me to integers modulo $1$. Is this correct, are the integers ...
2
votes
1answer
46 views

Proof of a Field Extensions Theorem

Consider the following result. Theorem : Let $E/F$ be a finite field extension of degree $n$ and let $V$ be a vector space over $E$. Then $$ \dim_F V = [E:F] \dim_E V. $$ Now, it seems like a ...
1
vote
1answer
44 views

Algebraic extension of rational numbers.

Let $1<m_1,\ldots,m_r\in{\mathbb{Z}}$. If $K=\mathbb{Q}(\sqrt{m_1},\ldots,\sqrt{m_r})$, and $1<n\in{\mathbb{Z}}$ so that $m_i\nmid{n}$. Is true that $\sqrt{n}\notin{K}$? Added: In addition ...
2
votes
2answers
29 views

A field $K$ is an algebra

I learned this definition of an algebra recently. The definition is: A vectorspace $V$ over a field $K$ is is an algebra if there exists $K$-bilineair map $\varphi\colon V\times V\rightarrow V$ which ...
1
vote
0answers
21 views

If $k$ is a field then $\text{End}_k(k^2)$ is simple

Let $k$ be a field. I have to show that $\text{End}_k(k^2)$ is simple. First of all, I don't see why this is true. For example, if $k=\mathbb{C}(x_1,x_2,\dots)$ then $\varphi\colon k^2\rightarrow ...
0
votes
2answers
35 views

Existence of finite field extension containing a root

I've been thinking about my previous question a bit more, and I'm afraid I still don't quite understand. See: Can the natural embedding $K\to K[X]/(f)$ be extended to form an isomorphism $L/K\to ...
1
vote
1answer
45 views

Calculating Splitting field

Find the splitting field of the polynomial and degree over $\mathbb{Q}$ $P(X)=X^4+2$. The roots of $P(X)$ are $\sqrt[4]{2}\sqrt{i},\ -\sqrt{i}\sqrt[4]{2}, \ i\sqrt{i}\sqrt[4]{2},\ ...
0
votes
0answers
19 views

Minimal polynomial (Field theory)

Let $\alpha$ be the real positive fourth root of 2. My question is: $Polmin(\alpha,\mathbb{Q}(i))=X^4-2$? Because $Polmin(\alpha,\mathbb{Q}(i)) | Polmin(\alpha,\mathbb{Q})=X^4-2$ But ...
2
votes
1answer
32 views

Galois group of order 2^4

Find the galois group the polynomial $f(X)=(X^2-2)(X^2-3)(X^2-5)(X^2-7)$ over $\mathbb{Q}$. A splitting field for $f(X)$ is $K=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7})$. We must have ...
2
votes
0answers
24 views

Finite dimensional field extension, finitely many intermediate fields

Good morning, My question is the following: Does every finite dimensional field extension have finitely many intermediate fields? I thought about it quite a while and know that the following is true: ...
1
vote
1answer
38 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 ...
1
vote
1answer
67 views

Is there a short proof of the formula for Legendre symbol $(\frac{2}{p})=(-1)^{(p^2-1)/8}$?

Let $p$ > 2 be a prime number. I found in wiki a complex proof for this Legendre symbol: $$\left(\frac{2}{p}\right) = (-1)^{\frac{(p^{2}-1)}{8}}$$ Can anyone give me a short solution please?
0
votes
2answers
43 views

Separable extension [closed]

Let $\alpha$ algebraic over $k$ of characteristic $p>0$ Prove that $\alpha$ is separable over $k$ if and only if $k(\alpha)=k(\alpha^p)$. Any suggestion, please.
6
votes
4answers
92 views

Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$

Is there any way to determine the Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$ not using the discriminant? Thanks!
1
vote
1answer
31 views

Can the natural embedding $K\to K[X]/(f)$ be extended to form an isomorphism $L/K\to K[X]/(f)$?

I'm studying for an abstract algebra exam (covering commutative rings and Galois theory). As an exercise, I'm trying to work out on my own a proof of the theorem that, given a field $K$ and a ...
-1
votes
1answer
76 views

There are no field structures on $\mathbb{R}^3$, but what of $\mathbb{R}^n$ for $n\geq 4$?

Has it been proved that there do not exist nice field structures on $\mathbb{R}^n$ for $n\geq 4$? The quaternions $\mathbb{H}$ fail due to lack of commutativity and the bicomplex numbers ...
3
votes
1answer
61 views

Proper Field extensions

Given a field $F$, is there a proper field extension $K$ such that any root in $K$ of a polynomial in $F[X]$ is in $F$? Note: I am not looking for the algebraic closure of $F$. One candidate is the ...
0
votes
3answers
37 views

Finding the order of elements in a Galois Field

Does there exist a Galois field GF(4)? GF(4)={0,1,2,3}; If we take this Galois field, then the element '2' is not having any degree..? So is it possible to construct GF(4) ?
0
votes
1answer
23 views

Separable polynomial and algebraic extension

If $f\in F[t]$ is separable and $E/F$ is an algebraic extension, then how can I be sure that $f$ is separable as an element of $E[t]$? I thought it is a trivial question...but now I think it is ...
8
votes
1answer
331 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
1
vote
0answers
26 views

Regarding nomenclature of a vector related to field automorphisms

Is there a particular designation in use for the following type of vector, constructed by taking a collection of basis elements $\beta_1$, $\beta_2$, $\dotsc$, $\beta_n$ for a field extension, ...
0
votes
2answers
88 views

Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
3
votes
2answers
65 views

In which Fields, does $x^n-x$ have a multiple zero?

In which Fields, does $x^n-x$ have a multiple zero? Attempt: Let $f(x) = x^n-x = x(x^{n-1}-1)$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. ...