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 topic....

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0
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0answers
15 views

What's an example of a non-normal, purely inseparable field extension?

Certainly every purely inseparable simple extension must be normal, since the minimal polynomial of the generating element $a$ must look like $X^{p^r}-a$, which splits in the extension, so the example ...
2
votes
1answer
30 views

Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer

I have two questions. Which integers are equal to the norm of some Gaussian integer? In general, how many solutions does$$\text{N}(a) = k$$have for a given $k \in \mathbb{Z}$? I am investigating the ...
3
votes
1answer
114 views

Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients of ...
5
votes
1answer
40 views

Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$? [duplicate]

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?
2
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2answers
79 views

Question in finding a new $\mathbb{Q}$-basis for $F/\mathbb{Q}$.

Let $F$ be the splitting field of $x^4 - 2$ over $\mathbb{Q}$. Let $G$ be its Galois group. When viewed as a $\mathbb{Q}$- vectorspace, $F$ has the following basis: $$\mathcal{B}=\{1,2^{1/4},2^{1/2},...
5
votes
0answers
42 views

Is an algebraic field extension $k \subseteq K$ normal if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$?

Over a perfect field $k$ it is well known that an algebraic field extension $k \subseteq K$ is normal if and only if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$, as ...
6
votes
3answers
74 views

Field with $125$ elements

I want to construct a field with $125$ elements. My idea is to consider the polynomial ring $\Bbb F_5[x]$. It is enough to find an irreducible polynomial $f\in \Bbb F_5[x]$ of degree $3$ because then $...
2
votes
1answer
104 views

Vakil's FOAG, Exercise 9.2.K: Transcendental Complex Numbers

How does one realize a transcendental complex number as a maximal ideal of $\mathbb{Q}(t) \otimes_{\mathbb{Q}} \mathbb{C}$? This is the essence of Exercise 9.2.K in Vakil's FOAG. Here is what I've ...
0
votes
3answers
46 views

Raising element of field to characteristic power

I came across this in a set of notes. Let $K$ be a field of characteristic $p$ and let $\lambda\in K$. Then $$\lambda^{p-1}=1.$$ I've never seen this before. Is it correct?
1
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2answers
50 views

Let $E=\mathbb{Q}(2^{1/3})$. What is the normal closure of $E/E$?

Let $E=\mathbb{Q}(2^{1/3})$. What is the normal closure of $E/E$? My thought is the $A(2^{1/3})$ where $A$ is an algebraic closure of $\mathbb{Q}$. But I am not sure whether it is correct and why... ...
2
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1answer
33 views

What does the notation $\overline{\mathbb R}$ mean in that context?

In an old question, it can be read that "the finiteness of $\text{Gal}(\overline{\mathbf R}/\mathbf R)$" is one of the "impressive finiteness results in mathematics". I commented the question to know ...
1
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0answers
22 views

Proof verification: Show that the fixed field is $\mathbb{Q}(\sqrt{3})$

Let $H$ be the subgroup $\{i,\alpha\}$ of $\text{Gal}_{\mathbb{Q}}\mathbb{Q}(\sqrt{3},\sqrt{5}),$ where $i$ is the identity map and $\alpha$ is defined as $\alpha(\sqrt{3})=\sqrt{3}$,$\alpha(\sqrt{5})=...
1
vote
1answer
24 views

Totally real Galois extension of given degree

Let $n≥1$ be an integer. I would like to prove (or disprove) the existence of a subfield $K \subset \Bbb R$ such that $K/\Bbb Q$ is Galois and has degree $n$. It is easy to construct such a subfield ...
5
votes
3answers
4k views

How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?

I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$. I am interested in counting how many such $...
38
votes
2answers
588 views

Shortest irreducible polynomials over $\Bbb F_p$ of degree $n$

For any prime $p$, one can realize any finite field $\Bbb F_{p^n}$ as the quotient of the ring $\Bbb F_p[X]$ by the maximal ideal generated by an irreducible polynomial $f$ of degree $n$. By dividing ...
1
vote
2answers
79 views

Splitting field of $x^9-x$ over $\mathbb{Z}_3$.

Let $F$ be a splitting field of $x^9-x$ over $\mathbb{Z}_3$. $1$. Show that there is an $\alpha \in F - \mathbb{Z}_3$ such that $\alpha^2=2$. $2$. Show that $\mathbb{Z}_3(\alpha)$ is a splitting ...
5
votes
1answer
31 views

Converse statement related to primitive element theorem

The classical proof of the primitive element theorem (over $\mathbb Q$) implies the stronger result that if $\alpha,\beta$ are two algebraic numbers over $\mathbb Q$, then ${\mathbb Q}(\alpha+t\beta)={...
-6
votes
1answer
76 views

Are these algebraic properties controversial? [on hold]

I wonder whether the following properties of an algebraic system controversial or are making damage to the algebraic system rendering it useless. We have two elements, say $w_1$ and $w_2$ with the ...
2
votes
1answer
29 views

What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
2
votes
2answers
86 views

Calculating Galois group of $\mathbb Q(\cos \frac \pi 8)/\mathbb{Q}$

What's a quick elegant way to compute the galois group of $\mathbb Q(\cos \frac \pi 8)/\mathbb Q$? I found the minimal polynomial to be $x^4-x^2-\frac 18$ but computing things directly is just ...
3
votes
1answer
31 views

Equivalent condition for $F[a^{\frac 1p}]=F[b^{\frac 1p}]$ for $\mathrm{char}F$ coprime to $p$

Let $p$ be prime and suppose $\mathrm{char}F$ is coprime to $p$ and $F$ contains the roots of unity. Why is it true that $F[a^{\frac 1p}]=F[b^{\frac 1p}]$ if and only if there's so $i$ coprime to $p$ ...
0
votes
0answers
12 views

Effect of seeds on the generation of keystream from a LFSR

I have a question regarding something I noticed about LFSR seeds. I tried different seeds in a simple LFSR with polynomial of “x4+x+1”, most cases I got equal amount of 1s and 0s in the keystream ...
4
votes
3answers
57 views

Slick proof $\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}p, \sqrt[\leftroot{-2}\uproot{2}p]{2}]/\mathbb Q)\cong S_p$?

After having seen a lengthy and painful calculation showing $\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}3, \sqrt[\leftroot{-2}\uproot{2}3]{2}]/\mathbb Q)\cong S_3$, I'm wondering whether there's a ...
1
vote
1answer
33 views

Hartshorne: Definition of $K^*$ where $K$ a function field of scheme.

Let $X$ be a noetherian integral separated scheme which is regular of codimension one. Let $K$ be the function field of $X$. Now let $f \in K^*$, (I am interpreting $K^*$ to be the set of field ...
1
vote
0answers
24 views

Separable but not reduced? [duplicate]

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every field extension $L/\Bbbk$, and reduced if its underlying ring is reduced. Separable always implied reduced, and I found ...
2
votes
4answers
70 views

Algebraic expression of Prime of form $4k-1$

Every prime of the form $4k+1$ can be written as an algebraic expresion of sum of two squares. Question: If $p=4k-1 $, can it be written as a sum of some powers? (algebraic exprssion like $p= y^3+ (...
2
votes
1answer
32 views

minimal polynomial in Kummer extension

Let $n>1$ be an integer. Let $K$ be a field such that $n$ does not divide the characteristic of $K$ and $K$ contains the $n$-th roots of unity. Let $\mu_n\subseteq K$ be the set of $n$-th roots of ...
1
vote
2answers
20 views

Is the following polynomial solvable in radicals?

Is the polynomial $x^8-x^6+2x^4-6x^2+1$ solvable in radicals over $\mathbb{Q}$? I am unsure how to solve this. I don't know how to compute the Galois group, and the discriminant seems much to hard to ...
0
votes
1answer
17 views

Field structure of non solvable field extensions

I was considering the base field $D$ which is some solvable extenstion of $ \mathbb{Q}$, and a polynomial that isn't solvable in radicals such as $ x^5 - x + 1$. If we let $\zeta$ be a root of this ...
1
vote
1answer
30 views

Confusion about generated subrings and subfields

In Milne's field theory notes he defines, given an extension $E/F$ and a subset $S\subset E$, the subfield of $E$ generated by $F$ and $S$ as the smallest subfield of $E$ containing both $F$ and $S$. ...
1
vote
1answer
24 views

Generic Element of Compositum of Two Fields [duplicate]

I'm interested in understanding compositum of general fields better. Assume we have $\Omega/K/F$ and $\Omega/L/F$ field extensions, and consider the composite $KL$. It seems to me that every $a \in ...
0
votes
1answer
19 views

Does any isomorphism of splitting fields automatically fix the base field?

I was wondering whether it's true that if $L_1,L_2$ are splitting fields of $f\in \Bbbk[x]$ over $\Bbbk$ then any isomorphism between them as fields must fix $\Bbbk$. If this is true, why? Is there ...
3
votes
0answers
58 views

Does the uniqueness of splitting fields up to non-canonical isomorphism fall from algebraic geometry?

Does the uniqueness of splitting fields up to isomorphism over the base field fall out from some deep results in algebraic geometry, or is it something special to fields which I shouldn't expect to ...
1
vote
0answers
17 views

$F(a_1,\ldots,a_n) = F(a_1+t_2 a_n+\cdots+t_n a_n)$ if Extension is Simple?

Suppose that F is an infinite field and that $L = F(a_1,\ldots,a_n)$ is a simple extension of $F$. Can you show, without the assumption that $[L:F] < \infty$ that $\exists t_2,\ldots,t_n \in F$ ...
3
votes
1answer
59 views

Is the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + t^2)$ irreducible over $k(t)$?

Let $k$ be an algebraically closed field of characteristic 2 and let $k(t)$ be rational function field of one variable. Consider the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + ...
1
vote
1answer
295 views

The elements in the composite field $FK$

Where $F$,$K$ are two fields. What does the element in the composite field $FK$ look like? All the elements are generated by the elements of $F$ and $K$? (combination of the elements of $F$ and $K$) I ...
0
votes
1answer
138 views

Would Euclid be satisfied by the construction of the 17-gon given by Gauss?

In our lecture on Algebra we were given the following exercice: Construct the regular 5-gon using straightedge and compass. (only using elementary geometric reasonig) If you construct the length ...
0
votes
0answers
28 views

Why $a+b$ is a generator of $F(a,b)$ over $F$, where $F$ is a field of characteristic zero.

Let $F$ be a field of characteristic zero. Assume that $a$ and $b$ are algebraic over $F$. The primitive element theorem says that there exists $w \in F(a,b)$ such that $F(a,b)=F(w)$; such $w$ is ...
2
votes
0answers
18 views

When the sum of two generators of a simple field extension is also a generator?

Let $F$ be a field of characteristic zero, and let $a$ be algebraic over $F$, so $K=F(a)$ is a finite separable field extension. Assume that $b$ is also a generator for $K$ over $F$, namely $K=F(b)$. ...
28
votes
7answers
2k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
-2
votes
0answers
17 views

field trace function is linear

Assume $E / F$ is a finite extension, the trace function is defined as $$\operatorname{Tr}(a_1)=[E:F(a_1)](a_1+a_2+\ldots+a_n)$$ (where $a_j$ are all of the roots of $\min(a_1,F)$). Then, what I want ...
8
votes
5answers
578 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
0
votes
1answer
50 views

why the algebra $A = k[x]/(x^n)$ has finite representation type? [closed]

Suppose that $k$ is algebraically closed. Then why the algebra $A = k[x]/(x^n)$ has finite representation type? please clarify the answer.
1
vote
1answer
22 views

Primitive element theorem w/o Galois theory (as in Lang's Algebra)

I want to understand PET in a form: every finite seaparable extension is simple. I don't want to use ideas from Galois theory (at least a biection between subgroups of Galois group and intermediate ...
2
votes
1answer
58 views

Solvability and reducibility of a polynomial in a “chain” of finite fields

This question is generalized based on my previous question: Is $x^5 + x^3 + 1$ irreducible in $\mathbb{F}_{32}$ and $\mathbb{F}_8$? Problem: Consider an irreducible polynomial $f = x^4 + x^3 + 1$ in ...
2
votes
1answer
63 views

What is $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p(X)} \mathbb{F}_p(\sqrt{X})$?

I am trying to understand what $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p (X)} \mathbb{F}_p(\sqrt{X})$ is. $\mathbb{F}_p(\sqrt{X})$ is the field of rational functions in $\sqrt{X}$. What is it ...
0
votes
0answers
32 views

does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
2
votes
2answers
41 views

Is there an infinite field F with char(F)=p and not algebraically closed field?

Is there an infinite field F with characteristic of the field $F$ is $p$ (p is prime) and not algebraically closed field ?
3
votes
3answers
282 views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
4
votes
1answer
50 views

Is $x^{2\cdot 3^n}+x^{3^n}+1$ irreducible (mod 2)?

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, it seems that the polynomials of the form $$x^{2\cdot3^n}+x^{3^n}+1 \pmod 2$$ are irreducible. ...