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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1answer
35 views

$\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$.

I need to show that $\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$. My approach was to find a bigger proper ideal containing $f(x,y)$ but i am unable to ...
3
votes
1answer
50 views

Elements of $GL_{2}(\mathbb{Z})$ of finite order

Prove that any element of $GL_{2}(\mathbb{Z})$ of finite order has order $1,2,3,4,6$ using FIELD THEORY. My idea is to reduce such a finite order matrix say $A$ with order $n$ to modulo a prime $p$. ...
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0answers
13 views

Trascendental extension [duplicate]

Let $p$ a prime number and let $\mathbf{F}_p(X)$ the field of rational functions over $\mathbf{F}_p$. The degree $[\mathbf{F}_p(X):\mathbf{F}_p(X^p)]$ is $p$? I'm a little confused; thanks.
2
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1answer
15 views

Square root of one of $2,3,6$ exist in a prime field

Using Gauss's law of Quadratic Resiprocity it is immediate that one of $2,3,6$ is a square in $\mathbb{F}_{p}$. I am looking for a solution which uses basic field theory only. I was thinking of ...
0
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3answers
155 views

Can we say “commutative ring = field”?

We know the difference between ring (R) and field (F) is that R does not guarantee multiplication is commutative. Now, if considering commutative R, which means (R,.) is a group, can we say: ...
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1answer
17 views

Let K/F be a transcendental extension , then does every F-homomorphism has to be an automorphism?

I have figured out that if $K/F$ be an algebraic extension , then does every F-homomorphism need not be an automorphism . But I can't figure it out for in thee trasncendental case.
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0answers
39 views

Find $[\mathbb{Q}(\sqrt [3] {3},\sqrt [3] {2}):\mathbb{Q}]$ [duplicate]

I am trying to find $[\mathbb{Q}(\sqrt [3] {3},\sqrt [3] {2}):\mathbb{Q}]$. My guess is it is $9$. There are 3 possibilities-3,6,9. If it is not 9 then $X^{3}-3$ is not irreducible over ...
3
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3answers
58 views

$[\mathbb{Q}(\sqrt [3] {2}+\sqrt {5}):\mathbb{Q}]$ [duplicate]

What is $[\mathbb{Q}(\sqrt [3] {2}+\sqrt {5}):\mathbb{Q}]$? A straight forward way will be to just set $x=\sqrt [3] {2}+\sqrt {5}$, take powers and reach at a polynomial in $x$ and show the polynomial ...
4
votes
1answer
69 views

$\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?

Is there a general or elegant way to approach this problem? One can show that $\sqrt 5+\sqrt[3] 2$ is a root of the hexic $x^6-6x^4-10x^3+12x^2-60x+17$, which should then be its minimal polynomial to ...
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0answers
24 views

Field isomorphism and order of elements

I know that group isomorphism preserves order of element but can someone plese tell me does field isomorphism preserves order of elements?
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0answers
81 views

Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
0
votes
2answers
33 views

every irreducible polynomial has a root in some field extension

We know the following fact from field theory. Let $F$ be a field and $p(X)$ an irreducible polynomial in $F[X]$. Then we can find a field extension $L$ of $F$ such that $p(X)$ has a root in $L$. ...
0
votes
1answer
28 views

Prove $\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$

I want to know why the following two are equivalent: $$\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$$, where $\mathbb{Q}$ is the rational number field, and ...
1
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2answers
33 views

Is $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ a splitting field of some polynomial

Is it true that the extension $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ is the splitting field of some polynomial over $\mathbb{Q}$? My guess is no. But I can not prove it. Some observations I made are as ...
0
votes
2answers
47 views

dimension of $\mathbb{Q}(\sqrt2)$

How can I prove that the splitting Field $\mathbb{Q}(\sqrt2)$ over the rational numbers $\mathbb{Q}$ is two dimensional vector space over $\mathbb{Q}$ ?
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1answer
20 views

Field Proofs with Multiplicative Inverses

I've been staring at these two for a while and I can't come up with anything concrete to start. Hints on how to begin would be greatly appreciated, full solutions are not necessary (and preferably ...
2
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0answers
11 views

Is the embedding problem with a cyclic kernel always solvable?

This question comes from this question by user72870. I shall explain how it relates to that question at the end. Let me shortly define my question: We call an embedding problem a diagram of the form: ...
1
vote
1answer
20 views

Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
0
votes
1answer
31 views

When people say, “K is an extension of k with dimension n”, do they mean as an algebra or as a vectorspace?

For instance, consider k(x), (the fraction field of k[x]). k(x) has dimension 2 as an algebra over k, but dimension \omega as a vectorspace over k. Which one are they talking about, and how can I ...
0
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2answers
58 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 ...
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votes
2answers
94 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
27 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
36 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
58 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
15 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
68 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 ...
-1
votes
0answers
28 views

Cubic resolvent of quartic.

Where does the cubic resolvent of quartic come from? I would like to know its derivation since I am having a hard time memorising the formulas.
3
votes
2answers
82 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
21 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
45 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
33 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
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0answers
20 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
37 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?
33
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?
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0answers
29 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
32 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
34 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
46 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
41 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 ...
1
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0answers
51 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
58 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
64 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
23 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
140 views

Give an example of a field where -1=1

The question is to find a counterexample to the following: In every field $F$, $-1$ is not equal to $1$. My intuition leads me to integers modulo $1$. Is this correct, are the integers modulo $1$ a ...
2
votes
1answer
42 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
42 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 ...