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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5
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1answer
50 views

I know Galois theory is used to study fields using properties of groups. Is it ever used to study groups using properties of fields?

More specifically, are there any results in pure, abstract group theory that are most easily proved using Galois theory?
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0answers
35 views

$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
0
votes
3answers
42 views

How can it be shown that for some prime p, $\mathbb{Q}$[$\sqrt{p}$, $\sqrt[3]{p}$] = $\mathbb{Q}$[$\sqrt[6]{p}$]?

I was told to consider the degrees but I'm not sure how the degrees of the polynomial so can help me here.
2
votes
0answers
20 views

On the restriction of field homomorphisms to subfields

Let $F/L/K$ be field extensions with $L/K$ finite. Let $H=\text{Hom}_K(L,F)$ be the set of field homomorphisms $L\rightarrow F$ that fix $K$. Take $\alpha\in L$, and let $\lbrace ...
1
vote
3answers
32 views

Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...
0
votes
2answers
42 views

how can one show that $\mathbb{Q}$($\sqrt{3}$, $\sqrt[3]{3}$, $\sqrt[4]{3}$, …) is algebraic but not finite dimensional?

The fact that this extension is infinite seems almost obvious and this is what makes it difficult to prove that the extension is algebraic. I would be able to do it for a finite case by identifying ...
0
votes
0answers
21 views

How to calculate the discriminant of a cubic equation easily

I'm trying to show the degree of the splitting field of a cubic polynomial with a zero quadratic term is related to the discriminant of the polynomial. On this process, i am trying to find the product ...
-1
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1answer
31 views

simple algebraic extensions with the same minimal polynomial [closed]

I can't see why $ji^{-1}$ is the identity on $K$, could someone explain please
1
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1answer
26 views

Finding a transcendence basis of a field over a rational function field in positive characteristic

Let $p$ be a prime number, and $k = \Bbb{F}_p(t)$ be a function field over $\Bbb{F}_p$. Let $R = k[x,y]/(x^p+y^p-t)$, and $K = \operatorname{QF}(R)$ be the quotient field of $R$. I need to find the ...
1
vote
1answer
39 views

How to determine the minimal polynomial of $\sqrt{3 + 2\sqrt{2}}$ over $\mathbb{Q}$?

I first let $\alpha = \sqrt{3 + 2\sqrt{2}}$ and $\alpha^2 - 3 = 2\sqrt{2}$. This gives us $(\alpha^3 - 2)^2 = 8$. Expand the polynomial we obtain that $x^4 - 6x^2 +1$ has $\sqrt{3 + 2\sqrt{2}}$ as a ...
2
votes
0answers
21 views

Quick help on why this extension is of degree $2$

The set up is, Splitting field $K=\mathbb{Q}(i,\alpha)$ where $\alpha=2^{\frac{1}{4}} \in \mathbb{R}$. The three obvious subfields of $K$ with degree $2$ over $\mathbb{Q}$ are..? Answer is ...
2
votes
0answers
21 views

Maximum ideal in field

Let $k$ be a field, $n ∈ \mathbb{Z}>0$, and $α_1, α_2, ..., α_n ∈ k$. Prove: $(x_1 − α_1, x_2 − α_2, \ldots, x_n − α_n)$ is a maximal ideal. I cannot figure out how to prove this; what is meant ...
0
votes
0answers
29 views

Why does a subfield of $\mathbb{R}$ not contain every real number?

Let $F$ be a subfield of $\mathbb{R}$. Since the completeness axiom holds in $\mathbb{R}$, it will also hold in $F$. But every real number can be represented as an infinite cauchy series of rational ...
2
votes
2answers
44 views

Can we use Eisenstein's Irreducibility Criterion to show that $x^4+1$ is not reducible in Q?

As such: Let $a(x)=x^4+1\in\mathbb{Q}\left[x\right]$. Then choose any prime $p$. By Eisenstein's Criterion, we see that $p\nmid 1$, $p\mid 0$ (since all coefficients of intermediate terms are 0), and ...
2
votes
1answer
53 views

Any method to solve this system of equation?

We have m variables $ x_{1},x_{2},...,x_{m} $ which are elements of field $F_{p}$ and we are given m equations of the form $$\sum_{i=1}^{m} x_{i}^{n} = c_{n} \mod p \qquad for \: 1 \le n \le m$$ ...
0
votes
1answer
37 views

Show $\sqrt[3]{2}\notin F$

Suppose that $F$ is the infinite extension of $\mathbb{Q}$ obtained by adjoining the square root of every integer (positive or negative). I'm trying to show that $\sqrt[3]{2}\notin F$. I have no idea ...
0
votes
1answer
40 views

Is $K^{n}$ Zariski Hausdorff when $K$ is a finite field?

Assume that $K$ is a finite field. Is it true to say that $K^{n}$ is a Hausdorff topological space with Zariski topology?
0
votes
1answer
31 views

How do I find a basis of $\mathbb{Q}(i,\sqrt{2}+i,\sqrt{3}+i)$ over $\mathbb{Q}$?

I can see that $[\mathbb{Q}(i):\mathbb{Q}]=2$ and that each of $[\mathbb{Q}(\sqrt{2}+i):\mathbb{Q}]$ and $[\mathbb{Q}(\sqrt{3}+i):\mathbb{Q}]$ is $4$. This implies that ...
1
vote
1answer
32 views

structure of extension by quadratic elements

Is it true that if $\sqrt{b}\not\in\mathbb{Q}(\sqrt{a})$ for $a,b$ not squares in $\mathbb{Q}$, then $Gal(\mathbb{Q}(\sqrt{a},\sqrt{b}))\cong Gal(\mathbb{Q}(\sqrt{2},\sqrt{3}))$? Im seeing a bunch of ...
1
vote
0answers
25 views

Galois group of degree 3 extension

For $d$ not a perfect cube in $\mathbb{Q}$, I want to compute $Gal(\mathbb{Q}(\sqrt[3]{d})/\mathbb{Q})$. I believe it has to be the trivial group, since it sends roots of the minimal polynomial ...
1
vote
1answer
27 views

Show that $\mathbb{Q}(\sqrt{2})$ is a field.

Proof: Since $\mathbb{Q}$ is a field, then $\mathbb{Q}$ is a domain. (Theorem: if $R$ is a domain, then $R[x]$ is a field.) By the theorem, $\mathbb{Q}[x]$ is a field. So, letting $x = \sqrt{2}$, ...
1
vote
1answer
32 views

Irreducible factorisation of polynomial over quotient field

Let $F=\mathbb{Z}_3[x]/<x^2+1>$. Factor $x^4+2$ into irreducibles in $F[x]$. I know that $F$ is a field since $x^2+1$ is irreducible. The usual way to find out that a polynomial is irreducible ...
0
votes
1answer
40 views

The Ring extension isomorphic to the field extension

Let $\alpha$ be algebraic over $F$, with $F(\alpha)$ the smallest field containing both $F$ and $\alpha$, and with $F[\alpha]$ the smallest ring containing both $F$ and $\alpha$. I want to show ...
0
votes
0answers
24 views

Center of matrices over a field [duplicate]

I'm trying to find the center of $\mathbb{M}_n(K)$ with $K$ a field. I know what the center would be if $K$ was a ring, but I think this isn't the same for a field $K$. In particular I'm trying to ...
0
votes
1answer
37 views

How to find the minimal polynomial of $\mathbb{Q}(3^{1/2}+5^{1/3})$ over $\mathbb{Q}$?

How would I find this polynomial algebraically? For example, if I wanted to find the minimal polynomial of $\mathbb{Q}(3^{1/2})$ over $\mathbb{Q}$, I would set $\alpha=3^{1/2}$, square both sides, ...
1
vote
0answers
16 views

Factoring Bivariate Polynomial which Vanishes on Curve

Let $K$ be an infinite field, and let $f \in K[x,y]$. Suppose $f$ vanishes on $x = y$. Show that $x-y$ divides $f$. I want to be able to use the division algorithm, so I first form the field of ...
1
vote
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
votes
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. ...
1
vote
1answer
19 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 ...
0
votes
1answer
29 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 ...
0
votes
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
108 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
votes
0answers
21 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 ...
1
vote
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 ...
1
vote
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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votes
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?
1
vote
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
vote
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
votes
1answer
46 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
vote
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
vote
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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vote
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 ...