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 ...

learn more… | top users | synonyms (1)

1
vote
1answer
26 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!
2
votes
3answers
28 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 ...
0
votes
1answer
71 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 ...
2
votes
1answer
52 views

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 ...
2
votes
1answer
22 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 ...
1
vote
1answer
22 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 ...
4
votes
3answers
43 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 ...
1
vote
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 ...
2
votes
1answer
34 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 ...
0
votes
0answers
20 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.
1
vote
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 ...
2
votes
3answers
37 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.
2
votes
0answers
27 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 ...
1
vote
2answers
65 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 ...
1
vote
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 ...
1
vote
1answer
58 views

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

Show that $x^4+2x^2+4$ is irreducible in $\mathbb{Q}[x]$.
0
votes
2answers
37 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 ...
1
vote
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 $k$. ...
1
vote
1answer
41 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 ...
3
votes
1answer
73 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
votes
1answer
56 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 ...
0
votes
0answers
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 ...
1
vote
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, ...
2
votes
3answers
122 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 ...
1
vote
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 ...
2
votes
3answers
88 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 ...
3
votes
1answer
62 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} ...
-1
votes
2answers
44 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 ...
1
vote
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)= ...
2
votes
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 ...
0
votes
0answers
58 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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 $ ...
0
votes
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 $2$), ...
2
votes
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 ...
3
votes
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 ...
3
votes
2answers
47 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 ...
3
votes
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
votes
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 ?
2
votes
0answers
62 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 $(0,1,+,\cdot)$?
0
votes
0answers
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 ...
0
votes
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 ...
1
vote
2answers
53 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 ...
0
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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}} ...
6
votes
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 ...