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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0answers
18 views

Condition for field density

Could we conclude that every ordered field (so with characteristic 0) is always dense in itself. I know that density is a topology concept, but given the field is ordered, defining dense will be: $$ ...
7
votes
1answer
60 views

$\mathrm{Aut}(\mathbb{Q}(\pi)/\mathbb{Q})=$?

Perhaps a silly question. I'm trying to understand trascendental field extensions, but I can't find a lot of instructive examples. Consider the extension $\mathbb{Q}(\pi)/\mathbb{Q}$. What is its ...
0
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0answers
28 views

Finding the minimal polynomial of an element over a field

If I am finding the minimal polynomial of an element $\alpha$ over a field $F$, all I need to do is: Find a (monic?) polynomial $f$ in $F[x]$ that vanishes at $\alpha$. Show that $f$ is irreducible. ...
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1answer
17 views

Algebraic number with conjugates having modulus 1

Suppose $\alpha$ is an algebraic number lying in a number field $K$ that is a normal extension of $\mathbb{Q}$. Suppose all the conjugates of $\alpha$ have absolute value 1. Prove or disprove that it ...
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1answer
28 views

$Q(a,i)$ is isomorphic to a quotient of $ \mathbb Q[X,Y]$

Let $a\in \mathbb C$ be a 3rd root of 2, i.e. $a$ has minimal polynomial $X^3-2$ over $ \mathbb Q$. Claim: $ \mathbb Q[X,Y]/(X^3-2,Y^2+1) \cong \mathbb Q(a,i)$ How do I see this, do I need to ...
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2answers
17 views

Field properties to ensure mean existence

I want to know if there is some properties that when satisfied by a field, we guarantee the existence of a "mean" of two scalars in that field. I can formulate my question as: is there any properties ...
6
votes
1answer
105 views

Minimal polynomial of root of unity over quadratic field

Let $p$ be an odd prime and consider the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$ and its quadratic subfield $\mathbb{Q}(\sqrt{\pm p})=:K$. I am interested in the minimal polynomial of a root of ...
0
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0answers
18 views

Decomposition of a polynom in a field extension

Let $K$ a field, $P \in K[X]$ irreducible of degree $n$, $L$ an extension field of $K$ with degree $m$ and $d=gcd(m,n)$. I want to show that for every $Q$ irreductible factor of $P \in L[X]$, ...
0
votes
1answer
11 views

composite of a separable extension and a purely inseparable extension

Let $K$ be a field and $L$ be an algebraic closure of $K$. Let $E,F$ be subfields of $L$ containing $F$ such that $E/K$ is separable and $F/K$ is purely inseparable. Let $x_1,\cdots,x_n$ be some ...
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2answers
33 views

Prove that Every element in $F(c)$ can be written as $r(c)$ for some $r(x)$ of degree $< n$ in $F[x]$.

Let $p(x)$ be an irreducible polynomial of degree $n$ over $F$. Let c denote a root of $p(x)$ in some extension of F. Prove that Every element in $F(c)$ can be written as $r(c)$ for some $r(x)$ of ...
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2answers
30 views

Find a minimal polynomial over $\mathbb{Q}(i)$

Show that the minimal polynomial of $\alpha =\sqrt{2} - i$ over $\mathbb{Q}(i)$ is $x^2 -2ix -3$. Is this similar to showing that the minimal polynomial over $\mathbb{Q}$, because I know the ...
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2answers
45 views

Minimal polynomial over $\mathbb R$ [closed]

Show that the minimal polynomial of $\sqrt{2} + i$ over $\mathbb R$ is $x^2 -2\sqrt{2}x+3$. How do I approach this problem?
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1answer
26 views

Is anyone familiar with the notation $\sigma|_M$?

I am looking at Ian Stewart's Galois Theory 4th Edition, and unsure about what the notation means. Here's the theorem that the notation is first seen, Suppose $L:K$ is a finite normal extension ...
0
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0answers
13 views

Reference request: field extensions

This is a standard reference request for field extensions, algebraic extensions, and the like. My class is covering ch13/14 of D&F, and I would appreciate both canonical references and online ...
2
votes
0answers
38 views

Proof that $\mathrm{Aut}(K\left[ \sqrt[m]{g \in K} \right])/\mathrm{Aut}(K) = \mathbb{Z}_w, w | g $

Let $K$ be an extension field of $\mathbb{Q}$. That is $$ K = \mathbb{Q}[r_1,\ldots,r_k ]$$ I am considering $\mathrm{Aut}(K)$ which is the group of field automorphisms of $K$ and I wish to show ...
1
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1answer
24 views

Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field.

Given Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field of $3$ elements then, $F$ is a field with $27$ elements. $F$ is separable but not a normal extension of $F_3$. The ...
0
votes
2answers
25 views

Finite dimensional division ring over an algebraically closed field

I know that an algebraically closed field $K$ cannot have an finite dimensional proper field extension. But can there be a division ring containing $K$ such that $[D:K]<\infty $.?
0
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1answer
45 views

Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$ [duplicate]

Question 1: For $n \in \mathbb{N}$ explain why $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ We must show that it is normal and separable. $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is normal since it ...
0
votes
1answer
60 views

How many fields are there between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_{15})$?

$\mathbb{Q}(\zeta_{15})$ is a field extension of $\mathbb{Q}$, where $\zeta_{15}$.I am trying to find the number of $i$ such that: $\mathbb{Q} \subset L_i \subset\mathbb{Q}(\zeta_{15})$ Is there a ...
1
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1answer
32 views

If $K$ is the algebraic closure of $F$, is $K$ also algebraically closed?

This comes from a question about the fundamental theorem of algebra. Is the algebraic closure of $\mathbb{C}$ implied by the fact that $\mathbb{C}$ is the algebraic closure of $\mathbb{R}$? More ...
1
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1answer
43 views

Is this a field in linear algebra?

Let $\alpha$ be a root of $x^4+4kx+1=0$ where $k$ is an integer. Is $\Bbb Q[\alpha]=\{a_0+a_1\alpha+a_2\alpha^2+a_3\alpha^3; a_i\in\Bbb Q\}$ a field? I find it is quite hard to see $\Bbb Q[\alpha]$ ...
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0answers
29 views

A field extension K/F is biquadratic if [K:F] = 4 and if $K$ is generated by the roots of two irreducible quadratic polynomials

Artin claims that every biquadratic extension has the form $F(\alpha,\beta)$,where $\alpha^2 = a$ and $\beta^2 = b$ with $a,b\in F$. I do not think this is true, just consider $\alpha = 1 + \sqrt ...
0
votes
1answer
33 views

Splitting fields being Galois

For a finite extension $K/F$, $K$ is Galois over $F$ if $\mid Aut(K/F)\mid=[K:F]$. Is the splitting field of any polynomial containing a separable factor Galois?
4
votes
2answers
30 views

The line $y = \sqrt2x$ + $\sqrt3$

Does the line $y = \sqrt2x + \sqrt3$ contain a point $(x,y)$ such that $x$ is rational and $y$ is rational? I looked at the line $y = \sqrt2x$ and proved it has just one such point $(0,0)$. I also ...
1
vote
1answer
54 views

is f(x) = 0 irreducible in $\mathbb{Z} /2 \mathbb{Z}$?

Let's say I have a polynomial like $$f(x) = 4x^2 +12x +28$$ when I reduce this with respect to mod 2; I end up with $0$. Can I say that zero is irreducible in ...
0
votes
1answer
19 views

For $f\in\mathbb{Q}[x]$, Gal($f)\subset S_n$ is a subset of $A_n$ iff $\Delta(f)$ is a square in $\mathbb{Q}^*$

Let $f\in \mathbb{Q}[x]$ a monic irreducible polynomial, and Gal($f$) be a subgroup of $S_n$. How do I prove that Gal($f$) $\subset A_n\iff \Delta(f)$ is a square in $\mathbb{Q}^*$? I know what ...
2
votes
1answer
32 views

intermediate Field extension by irreducible polynomial

Thanks for any help or comments. Suppose $f(x)\in F[x]$ is an irreducible polynomial of composite degree $n$. So there exist an closure field $\bar{F}$ such that $f(x)$ is completely reducible in ...
2
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1answer
33 views

On the action of galois groups in towers of fields

I would like some confirmation on certain statements I believe to be true: Let $K\subset L\subset M$ be a tower of fields such that both extensions $L/K$ and $M/K$ are galois. Let $f(x) \in K[x]$ be ...
2
votes
1answer
41 views

Show that the degree of any irreducible factor of $x^8-x$ over $\mathbb Z_2$ is $1$ or $3$

Statement Prove that the degree of any irreducible factor of $x^8-x$ over $\mathbb Z_2$ is $1$ or $3$. The hint in the back of the book makes a certain claim that I wasn't sure about. The author ...
6
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0answers
76 views

If $E/F$ is finite, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically closed?

I'm struggling to understand the claim that if $E/F$ is a finite field extension, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically ...
0
votes
2answers
46 views

Galois group of the splitting field of the minimal polynomial over $\Bbb{Q}$

Determine the Galois group of the splitting field of the minimal polynomial of the following algebraic numbers $\sqrt{2}+\sqrt{3}+\sqrt{6}$ over $\Bbb{Q}$. It is clear that ...
6
votes
3answers
281 views

A field has only one isomorphic subfield to itself?

Let $E$ be a field and $F$ be a subfield of $E$ which is isomorphic to $E$. Then is $F$ equal to $E$? It seems to be clear but I couldn't prove it. Could you please explain this statement?
0
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1answer
41 views

Complex numbers of degree less than or equal to 2 over rational numbers

Im trying to figure out which complex numbers have degree $\leq$2 over $\mathbb{Q}$ and then figure out which have degree $\leq$2 over $\mathbb{R}$. For the first question, I know that it is at least ...
0
votes
0answers
26 views

Showing fields are algebraically closed

Let K be a field, and let P be separable and irreducible over K. Let L be a splitting field of P over K. I want to show that the fields K(u) and K(v) are isomorphic, where u and v are roots of P, ...
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1answer
40 views

Prove that $\mathbb{F}_p(x,y)/\mathbb{F}_p(x^p,y^p)$ is not a simple extension [closed]

Prove that $\mathbb{F}_p(x,y)/ \mathbb{F}_p(x^p,y^p)$ is not a simple extension by explicitly exhibiting an infinite number of intermediate subfields .
1
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1answer
33 views

Roots of polynomial in field extension

My question is stated below: Let $K=\mathbb Z_3 [x]$ and $p(x) ∈ \mathbb Z_3 [x]$ be defined by $p(x) = x^4 + x + 2$. Consider the field extension $\mathbb Z_3 [x]/(p(x))$. Define $q(x) ∈ \mathbb ...
22
votes
7answers
2k views

How to prove that a complex number is not a root of unity?

$\frac35+i\frac45$ is not a root of unity though its absolute value is $1$. Suppose I don't have a calculator to calculate out its argument then how do I prove it? Is there any approach from ...
2
votes
0answers
21 views

Transcendental extension not isomorphic to its closure

Suppose I'm given a field extension $K/F$ with $\alpha\in K$ transcendental over $F$, the claim is that $F(x)\cong F(\alpha)$. It's a statement without proof in our class notes, and the remarks ...
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1answer
21 views

Potentially diagonal $K$-algebra

In our abstract algebra class, we had two lectures on field extensions. Then we were given the following homework problem (with this problem on diagonal $K$-algebra). Let $K$ be a field. A ...
3
votes
2answers
49 views

Find the minimal polynomial of $\sqrt{3} +i$ over $Q(i)$ and over $Q(\sqrt{3})$

Find the minimum polynomial of $\sqrt{3} +i$ over $Q(i)$ and $Q(\sqrt{3})$. My solution: Over $Q(i)$ : Suppose $a = \sqrt{3} +i$, $a-\sqrt{3} =i$; so, the minimal polynomial is $x-\sqrt{3} ...
1
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2answers
41 views

Showing that every polynomial over the Algebraic Numbers has a $0$ in the Algebraic Numbers. [duplicate]

Let $\mathbb{A}$ denote the field of Algebraic Numbers: the field of all complex numbers that are algebraic over $\mathbb{Q}$. Assuming that every polynomial over $\mathbb{C}$ has a $0$ in ...
0
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1answer
28 views

Cyclic extension of degree $2$ of $\mathbb{F_2}$

I want to find a cyclic extension of degree $2$ of $\mathbb{F_2}$ The degree of its minimal polynomial is $2$, i.e. a quadratic polynomial. The only cyclic field of degree $2$ which I can think of ...
2
votes
2answers
89 views

Geometric reasons finite fields have prime power orders?

All variations of proofs that finite fields have prime power orders have a very algebraic feel to them. I was wondering - is there a more geometric way to see why this is true?
1
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3answers
93 views

Can every function be represented as polynomial [closed]

Can every function $f: R^n \to R^n$ or$R^n \to R$ be represented as polynomial either of degree $n$ or infinite degree. Are there any proofs to this statement if it is true?If it has no Taylor-Power ...
0
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1answer
31 views

Normal closure $L$ of $\mathbb{Q}(\sqrt[4]{7})$ and structure of $Gal(L/\mathbb{Q})$

I think I have done (a) but I need some guidance on (b), if possible (a). Find a normal closure $L$ of $\mathbb{Q}(\sqrt[4]{7})$ To construct the normal closure I could adjoin the roots of ...
12
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0answers
130 views

Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...
2
votes
2answers
54 views

$K= \mathbb{F}_2(\alpha)$ $\alpha$ root of $X^4+X+1 \in \mathbb{F}_2[X]$. Find degree and minimal polynomial

Question 1: Find $[K:\mathbb{F_2}]$ Idea: I have tried looking at the irreducibility of the polynomial, $X^4+X+1 $ and have so far been unsuccessful. Is there another way to do this apart from using ...
1
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1answer
19 views

Principal Ideals

Let $R$ be a commutative ring with unity. I'm trying to prove if every ideal of $R[X]$ is a principal ideal, then $R$ is a field. So it's sufficient to show $R$ is a division ring. Question: What ...
0
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0answers
22 views

is it true that closures preserve isomorphisms [duplicate]

Suppose I have two isomorphic integral domains $A$ and $B$. Are the fields of fractions of these two rings isomorphic as well?
1
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1answer
87 views

Prove that $\mathbb Q[\sqrt[3]2]$ is a field

We define the set: $$\mathbb{Q}[\sqrt[3]2]=\{a_{0}+a_{1}\sqrt[3]{2}+a_{2}\sqrt[3]{2^{2}}:a_{0}, a_1,a_2\in\mathbb{Q}\}$$ It's easy to prove all the properties of fields, except for the unit ...