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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$V$ 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
21 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
57 views

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

Show that $x^4+2x^2+4$ is irreducible in $\mathbb{Q}[x]$.
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33 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
14 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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30 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
71 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 ...
2
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1answer
55 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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13 views

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
115 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
72 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
61 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
42 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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0answers
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
51 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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0answers
53 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
32 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
49 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
49 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
46 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 ...
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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 ?
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61 views

$\overline{\mathbb Q}$ and $\mathbb C$ same first order theory

How do you show that $\overline{\mathbb Q}$ (the algebraic closure of $\mathbb Q$) and $\mathbb C$ have the same first order theory over the signature $(0,1,+,\cdot)$?
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17 views

Linear algebra over a division ring

As is well known much of the theory of matrices over a field $\mathbb F$ remains correct for matrices over a division ring $D$,(the main exception is the theory of determinants). In which books are ...
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2answers
38 views

Fixed field of a subgroup of a Galois group

For the Galois group $Gal(\mathbb{Q}(\sqrt2, \sqrt3, \sqrt5)/\mathbb{Q})$, I'm trying to understand how to find the permutations of the roots and how the subgroups of the Galois group are related ...
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2answers
52 views

Show that $F[x] / (x^2-1) \cong F \times F$ iff $char(F) \neq 2$

Let $F$ be a field. Show that $F[x] / (x^2-1) \cong F \times F$ iff $char(F) \neq 2$. Here is my work this far: If $char(F) = 2$, then $x^2-1 = (x-1)^2$, and hence $F[x]/(x^2-1)$ has a non-zero ...
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1answer
20 views

Problem involving a tower of fields with an algebraic and a normal extension

I seem to be stuck on the following problem about field extensions from an old prelim exam in algebra. Let $K$ be an algebraic field extension of a field $F$ and let $L$ be a subfield of $K$ such that ...
2
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1answer
30 views

Field extensions and minimal polynomials

Let $L/K$ be a field extension and $\alpha,\beta \in L$. Let $f\in K[X]$ be the minimal polynomial of $\alpha$ and $g\in K[X]$ be the minimal polynomial of $\beta$ Show the following: $$f \text{ is ...
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1answer
17 views

What is a requirement for an order of algebraic number field $K$ to be integrally closed domain?

Suppose there is an order $O$, a subring, of an algebraic number field $K$. What is needed (necessary and sufficient condition) for $O$ to be integrally closed domain? Or if we need to impose ...
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0answers
19 views

Does $\mathbb{Z}$ in ring of integers $O_K$ in number field $K$ have to be generated by $K$'s multiplicative identity $1_K$?

As title says, does $\mathbb{Z}$ in ring of integers $O_K$ in number field $K$ have to be generated by $K$'s multiplicative identity $1_K$? Or can there be multiplicative identity element ...
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1answer
33 views

Field Characteristic Is Prime…?

Consider the article: http://mathworld.wolfram.com/FieldCharacteristic.html It is stated that given a field and its multiplicative identity $I_{\times}$ that either: $$ \sum_{i=0}^{k}{I_{\times}} ...
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0answers
37 views

why similarity over $\bar{\mathbb{F}}$ of $A,B\in M_n(\mathbb{F})$ implies similarity over $\mathbb{F}$?

A clasic problem in linear algebra is to determine if two matrices $A,B\in M_n(\mathbb{F})$ are similar one to another. When $\mathbb{F}=\bar{\mathbb{F}}$, we know that $A,B$ are similar if and only ...
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1answer
72 views

Automorphisms of $\mathbb{C}(X)$ and their fixed field

I'm stuck at the very beginning of an exercise I have to do for my algebra class. We're looking at the field of $\mathbb{C}(X)$ and it's automorphisms. Let $a \in \mathbb{C}^*$, $ b \in \mathbb{C} $ ...
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3answers
351 views

Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
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0answers
77 views

Transcendental extension and algebraic closure

Let $t$ be a transcendental element over field $K$, and let $F/K(t)$ be finite extension. Assume that $E$ is algebraic closure of $K$ in $F$. How to prove that $E/K$ is finite and $[E:K] \mid ...
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1answer
37 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
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1answer
39 views

Prove that factor rings are fields [closed]

Prove that factor rings $\mathbb{Z}_3[x]/(x^3 + x^2 +2)$ and $\mathbb{Z}_3[x]/(x^3 -x +1)$ are fields, and these fields are isomorphic.
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19 views

Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
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2answers
76 views

$\mathbb Q(\sqrt2) \not \cong \mathbb Q(\sqrt[3]{2})$ [closed]

Prove that all fields $\mathbb Q(\sqrt2)$ and $\mathbb Q(\sqrt[3]{2})$ are not isomorphic
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1answer
21 views

What is the Characteristic Polynomial of an element over a field in this case?

Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and ...
1
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1answer
33 views

Intermediate fields of $X^p - 2 $

I've been working on an exercise I have to do for my algebra course. Exercise: Let p be prime and $L$ the splitting field of $ f = X^p - 2$ over $\mathbb{Q}$. a) Show that $ Gal(L/\mathbb{Q})$ is ...
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2answers
56 views

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomials over Q for any natural number n.

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomial over $\mathbb{Q}$ for any natural number n. This is an extended version of my previous question here. I know if $n=4$, ...
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3answers
103 views

Field extensions with(out) a common extension

Let $K$ be a field having two field extensions $L\supseteq K$ and $M\supseteq K$. Does there exist a field $N$ along with embeddings $L\to N$ and $M\to N$, such that the diagram $$ \require{AMScd} ...
3
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1answer
65 views

Find all the fields between $\mathbb{Q}$ and the splitting field of $x^4 + 81$

Let $f(x)=x^4+81 \in \mathbb{Q}[x]$. Find the splitting field $E$ of $f(x)$ and the extension degree $[E:\mathbb{Q}]$. Find all the fields $L$ with $\mathbb{Q} \leq L \leq E$. Are the roots of ...
2
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0answers
72 views

Show that $x^4+n(2-n)x^2+n^2$ reducible over Q for any natural number n.

Show that $$x^4+n(2-n)x^2+n^2$$ reducible over Q for any natural number n. Here is what I did $$x^4+n(2-n)x^2+n^2=x^4+2nx^2-n^2x^2+n^2=x^4+2nx^2+n^2-n^2x^2$$ $$=(x^2+n)^2-(nx)^2$$ ...