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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Find the irreducible polynomial of $\zeta_{12}$ and $\zeta_9$ over the field $Q(\zeta_3)$.

Let $\zeta_n=exp(2 \pi i/n)$ Find the irreducible polynomial of $\zeta_{12}$ and $\zeta_9$ over the field $Q(\zeta_3)$. Here I can find polynomial satisfied by these elements over $Q(\zeta_3)$ but i ...
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1answer
45 views

Which algebraic intuition can be used in fields

I wonder what basic laws of arithmetic of reals e.g. $x^n y^m = (xy)^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I ...
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11 views

Evaluating minimal polynomial over a field $F$ as a characteristic polynomial for a $F$-linear map.

I'm considering $K/F$ to be an extension of degree $n$. I've shown that for any $a\in K$, the map $\mu_a : K → K$ defined by $\mu_a(x) = ax$ for all $x\in K$, is a linear transformation of the ...
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28 views

Proving that the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ is irreducible modulo $p$ for every prime $p$. [duplicate]

I've computed the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ to be $x^4+10x^2+1$. I want to show that this polynomial is irreducible module $p$ for every prime $p$. How do I do ...
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44 views

Cyclotomic polynomials and Galois groups

According to this question I want to extend the question from there. Lets consider again the galois extension $\mathbb Q(\zeta)/\mathbb Q$ where $\zeta$ is a primitive root of the $7^{th}$ cyclotomic ...
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1answer
39 views

Spaces vs. Structures

Examples of spaces I've come across include vector spaces, inner-product spaces, and metric spaces. Examples of structures I've met include rings, fields, and groups. I have always understood spaces ...
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1answer
97 views

Cyclotomic polynomials and Galois group

Let $\zeta\in \mathbb C$ be a primitive $7^{th}$ root of unity. Show that there exists a $\sigma\in \operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ such that $\sigma(\zeta)=\zeta^3$. I already know ...
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20 views

The most general splitting of a field extension

Take $L/K$ an extension of the field $K$. I have questions on how we can "split" the extension $L/K$. (1) One knows that $L$ has a transcendence basis $(x_i)_{i\in I}$ over $L$, so that if $E := ...
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1answer
26 views

$K-$rational solution of the equation - Is $\mathbb{Q} \leq \mathbb{Q}_p$?

Let $P(x, y) \in \mathbb{Q}[x, y]$. We consider the equation $P(x, y)=0$. If $a, b \in \mathbb{Q}$ such that $P(a, b)=0$ then $(a, b) \in \mathbb{Q}^2$, is called a rational solution. If $K$ a ...
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1answer
28 views

Primitive element and field extension

If $K$ is an extension of field $F$ such that $[K:F]$ is finite and for two subfields $K_1$ and $K_2$ which contains $F$, either $K_2\subset K_1$ or $K_1\subset K_2$, then $K$ has a primitive element ...
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1answer
31 views

Why is $\sqrt{5}$ an element of every field of order $p^{2 e}$?

This was claimed in an answer to another question I asked but it's unclear to me why it's true. I'd also be happy with a reference that explains it. Thanks!
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3answers
34 views

$x^2+3$ has two zeros over ${\Bbb F}_p$ provided that $x^2+x+1\in{\Bbb F}_p[x]$ has two?

The following is an exercise in abstract algebra: If $p=1\pmod{3}$, then $x^2+x+1\in\Bbb{F}_p[x]$ has two zeros. Prove in this case that $-3$ is a quadratic residue mod $p$. Showing that ...
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1answer
78 views

Theorem about equivalent norms.

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be equivalent norms on a normed field. Then (i) $\|x\|_1<1$ iff $\|x\|_2<1$; $\|x\|_1>1$ iff $\|x\|_2>1;$ (ii) $\|x\|=1$ iff $\|x\|_2=1$. I want to ...
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1answer
101 views
+50

Galoisgroup of a polynomial

Task: Determine the galois group of $x^4-4x^2-11$ over $\mathbb Q$ The roots are obviously $\pm\sqrt{2\pm\sqrt{15}}$ But I have problems in checking whether just $\sqrt{2+\sqrt{15}}$ is generating ...
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1answer
24 views

Generated field is the same as composite field

I am considering Exercise 13.2.6 from Dummit & Foote's Abstract Algebra: Prove directly from the definitions that the field $F(\alpha_1, \alpha_2, \dots, \alpha_n)$ is the composite of the ...
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1answer
23 views

$F\subset L \subset F[a]$ fields, prove L is created by $f_a$

Let $F\subset L \subset F[a]$ be fields, and $f_a(x)=\sum_{k=0}^n a_k x^k$ is the minimal polynomial of $a$ over L. prove that $L=F\left[a_0,...,a_n\right]$. I've tried a few ways but I can't manage ...
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3answers
58 views

Computing the galois group of $x^3+3x+1$

I want to calculate the galois group of the polynomial $x^3+3x+1$ over $\mathbb Q$. And I am struggling in finding the roots of the polynomial. I only need a tip to start with. Not the full solution ...
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2answers
22 views

If $f$ is irreducible over a perfect field, then $f$ has no multiple zeros.

If $f (x)$ is an irreducible polynomial over a perfect field $F$, then $f (x)$ has no multiple zeros$(1)$. But I have also $2$ theorems, which are contradictory to the statement above ...
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1answer
39 views

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R.

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R. My attempt: Let $a \in S$. Then $1_S*a = a$ and this a ...
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23 views

Prove: given a ring R with left identity $e_l$ and the right identity $e_r$, then $e_l = e_r$. Another way to prove?

Suppose a ring R has the left identity ($e_l$) and the right identity ($e_r$). Then $e_l = e_l*e_r = e_r$. I was wondering if there's another way to do it. Thank you.
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1answer
31 views

Why is it not a sufficient condition to conclude that a is a unity based only on the information that $xa = x$ for all $x$ in $R$?

We have a ring $R$ as follows: Why is it not enough to conclude that $a$ is a unity if $xa = x$ for all $x$ in $R$? Is it because it is by definition that the unity satisfies $ax = xa = x$ for all ...
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3answers
39 views

How to show the isomorphism

I'm not sure how to show this $$\mathbb{Z}_3[X]/(x^3 -x +1)\cong\mathbb{Z}_3[X]/(x^3 -x^2+x +1).$$ Any help or hint would be helpful.
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0answers
30 views

Etymology of normal extensions and subgroups

According to wikipedia, a normal extension is a splitting field of a family of polynomials, and a normal subgroup is one that is invariant under conjugation. Why are normal extensions and normal ...
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2answers
66 views

$V$ is finite dimentional over field $K\iff$ field extension $L/K$ is finite

Let $L/K$ be a field extension and $V$ a non-zero vector space over $L$. Prove that: $V$ is finite dimensional over $K\iff V$ is finite dimensional over $L$ and $[L:K]<\infty$ for the first ...
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1answer
23 views

All automorphisms of splitting fields

Let $M \le \mathbb{C} $ be the splitting field of polynomial $ f(x) \in \mathbb{Q}[x] $. Find all automorphisms of field $ M $ in cases: 1) $ f(x) = x^6 - 1 $ 2) $ f(x) = x^{2011} - 1 $ 1) In ...
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1answer
58 views

Irreducible Polynomial in $\mathbb{Q}[x]$ [closed]

Show that $x^4+2x^2+4$ is irreducible in $\mathbb{Q}[x]$.
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40 views

Computing a Galois group

Let $K=\mathbb Q$ and $L$ be the splitting field of the polynomial $f(x)=x^4-6x^2+4$. I want to calculate the Galois group $\text{Gal}(L/K)$. The zeros of $f$ are obviously ...
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1answer
15 views

Proposition about of Purely inseparable extension

In the Book's Algebra, Lang, page 251. Proposition: Let $E^p$ denote the field of all elementos $x^p$, $x\in E$- Let $E$ be a finite extension of $k$. If $E^pk=E$, then $E$ is separable over $k$. ...
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1answer
43 views

Question about polynomials in finite fields

No doubt I'm missing something obvious here (my finite field theory is quite rusty) but I'm reading a book that claims that 1) $f(x) = x^2 - x - 1$ is irreducible over $F_q$ where $q = p^e$ for an ...
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1answer
77 views

Irreducible polynomials over the reals

Everybody knows that the degree of irreducible polynomials over the reals is either one or two. Is it possible to prove it without using complex numbers? Or without using fundamental theorem of ...
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1answer
58 views

Galois theory with the extension $\mathbb Q(\alpha)/\mathbb Q$

Let $\mathbb Q(\alpha)/\mathbb Q$ be an algebraic exntension with $\alpha=\sqrt{2+\sqrt{2}}$. 1) Show that the extension is a galois extension (normal and separable) 2) Show that $Gal(\mathbb ...
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What is the Frobenius element (if any) of this group.

For $F = \mathbb{Q}[\sqrt{5}]$ with $p = 2$ and prime ideal over p of $ q = (2, 1 + \sqrt{5})$ with the Frobenius element defined as $$ x^{Frob_q} \equiv x^p (mod q) $$ with $Frob_q \in Gal ...
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1answer
57 views

Roots of unity in splitting field of all polynomials of given degree

Let $K$ be the splitting field of all polynomials of degree $4$ in $\mathbb{Q}[x]$. For which $n\in \mathbb{N}$ the $n$-th primitive root of unity $\xi_n\in K$ ? I've shown that $K$ is an algebraic, ...
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3answers
126 views

Field homomorphisms of $\mathbb{R}$

How to prove that $\mathrm{Hom}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$ ? (We treat it as field homomorphisms. ) I know that $\mathrm{Aut}(\mathbb{R})=\{\mathrm{id}\}$ and ...
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1answer
17 views

Subfield generated by a multiplicative subgroup of the field

Let $F$ be a field with prime $p$ characteristic and let $X$ be a periodic subgroup of $(F,\, \cdot)$. Let now $K$ be the subfield of $F$ generated by the elements of $X$. Is it possible to describe ...
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3answers
93 views

Writing elements of field extension in terms of the basis determined by a root of a polynomial

Let $\alpha \in \mathbb{C}$ be a root of the irreducible polynomial $$f(X) = X^3 + X + 3$$ Write the elements of $\mathbb {Q}(\alpha)$ in terms of the basis $\{1, \alpha, \alpha^2\}$. The first ...
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1answer
65 views

The Kahler differential is zero

While studying for my commutative algebra exam, I have come across this problem. Let $K$ be a field. Let $A$ be a $K-$algebra, finitely generated as a $K$-vector space. Prove that $\Omega^1_{A/K} ...
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2answers
45 views

Do minimum polynomials always have a nonzero discriminant?

Let $f(x)$ be a minimum polynomial with integer coefficients. Does $f(x)$ Always have its discriminant equal to nonzero ? If so , why can't $f(x)$ have a repeated root ? For all clarity Im talking ...
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27 views

Mistake in proof that a polynomial $f$ irreducible in $F$ is irreducible in $E$ if $\gcd(\deg f, [E:F])=1$

This is a problem in James Milne's text on Galois Theory: Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $E$ be a field extension of $F$ with $[E:F] = m$. If $\gcd(m,n)= ...
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2answers
52 views

Galois-theory - a question about the galois group

I am dealing with galois theory at the moment and I came across with an example in the lecture and I got a question: Let $K=\mathbb Q$ and $L=\mathbb Q(\sqrt{2},\sqrt{3})\subset \mathbb C$. Lets ...
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59 views

How cannot localization of any integral domain respect to maximal ideal not be integrally closed?

Suppose that there is integral domain $I$. Now we take localization $I_m$ of $I$ respect to its maximal ideal $m$. $I_m$'s elements will consist of $a/b$ where $a \in I$ and $b \in m$. But integral ...
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1answer
48 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
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0answers
21 views

Extension Fields, complex numbers.

I have a question about the complex numbers given as an extension field. I know that complex numbers can be seen as $\mathbb{R}\left[x\right]/<x^2+1>$ (Well, I really don't know it yet, because ...
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1answer
34 views

Embeddings of a subfield of $ \mathbb{C} $

I'm trying to understand / solve the following problem: Let $ L \subset \mathbb{C} $ be a field and $ L \subset L_1 $ its finite extension ($ [L_1 : L] = m $). Prove that there are exactly $ m $ ...
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1answer
51 views

Why 2*3=6 in finite field?

In a finite field $F$, say of order 7. Suppose I didn't known that it must isomorphic to $\mathbb{F}_7$ By the definition of field, there exists $1 \in F$, and hence exists $1+1$ (denoted by $2$), ...
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2answers
53 views

Embed finite field in algebraic closed field

Let $\Omega$ be an algebraic closed field of characteristic $p$, then for any field $F$ of $q$ ($=p^a$) elements, $F$ can be embedded in $\Omega.$ I need above property to proof that every ...
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1answer
48 views

Does there exist $\mathbf{Q} \subset R \subset \mathbf{C}$, $R$ ring & not field

I am looking for an example of a field extension $k \subset F$ and a unital ring $R$ that is not a field such that $$k \subset R \subset F.$$ I know if $F$ is algebraic over $k$, then this is not ...
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2answers
48 views

Galois group, algebraic closure over maximal extension

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $\alpha \in \overline{\mathbb{Q}}\setminus \mathbb{Q}$ and let $K \subset \overline{\mathbb{Q}}$ be a maximal extension of ...
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0answers
31 views

Show that $K(\alpha,\beta)/K$ is simple

Let $L=K(\alpha,\beta)$ be an algebraic field extension, with $\alpha$ separable over K. Show that $L/K$ is simple. My attempt: If we could show that $L/K$ is finite and separable then the claim ...
2
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0answers
16 views

Connection between field and the frobenius homomorphism [duplicate]

Let K be a field with char(K)>0. How do I prove that every algebraic extension of K is a separable extension if and only if $\phi:x \rightarrow x^p$ is surjective ?