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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28 views

Field $F$ with $\operatorname{char}F=3$ and algebraic over $\mathbb{F}_3$ has a primitive root of unity.

Suppose that $F$ is a field with $\operatorname{char}F=3$ and $F$ is an algebraic extension of $\mathbb{F}_3$. Prove that $F$ contains a primitive $n$th root of unity for some $n>2$.
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1answer
40 views

A polynomial's irreducibility in $\Bbb{Z}_p$

Show that if $f$ is irreducible in $\Bbb{Z}_p[x]$ then $f$ divides $x^{p^n} - x$ for some $n \in N$. I know that: $f$ is irreducible, so $F = \Bbb{Z}_p / {\left\langle f\right\rangle}$ is a ...
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1answer
41 views

The characteristic of real-closed fields is zero?

We know that $F$ is a real-closed field if $F$ is not algebraically closed but $F(\sqrt{-1})$ is algebraically closed. So I have this question What can we say about $\operatorname{char}F$? Is it ...
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2answers
42 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
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1answer
35 views

If $F$ is a field, then any two algebraic closures are isomorphic by an isomorphism that is the identity on $F$.

To start, suppose $K_1$ and $K_2$ are two algebraic closures of $F$. (a) Let $P$ be the set of partial functions $f$ from $K_1$ to $K_2$ with the following properties: $F$ is contained in ...
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votes
1answer
26 views

Do real quadratic fields with unique primary factorization exist?

Bumped in Stillwell's book "Elements of Number Theory" into "The real quadratic fields with unique prime factorization are still not known ...". But doesn't $\mathbb{Q}[\sqrt{2}]$'s ring of integers ...
3
votes
1answer
71 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
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0answers
24 views

Question on primitive element for extensions of finite fields [duplicate]

I was given this question in field theory class which I am stumped on as after some effort I still cannot tackle. I am given a general natural number $n>1$ and am asked to prove or disprove: ...
2
votes
1answer
35 views

Another abstract algebra/field theory question

Suppose that $F$ is a field, $S \subseteq F^n$ and $I$ is an ideal in $F[x_1, \cdots, x_n] = F[\bar{x}]$. Define $$I(S) = \{ f \in F[\bar{x}]: f(\bar{s}) = 0, \forall \bar{s} \in S\}$$ and ...
4
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1answer
38 views

splitting field of $x^8-1$ over $\mathbb F_3$

Suppose $F=\mathbb F_3$ and $f(x)=x^8-1$ in $F[x]$. I tried finding the Galois group of the splitting field of $f(x)$ over $F$ and I'm not so sure if what I did was correct. I began by looking at ...
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0answers
27 views

Subgroups of the multiplicative group of a Field [duplicate]

I am reading S. Roman's book "Field Theory". In chapter $1$ I found the following exercise Let $F^*$ be the multiplicative group of all nonzero elements of a field $F$. We have seen that if G is a ...
4
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2answers
53 views

primitive element $a$ of $\mathbb F_{p^n}/\mathbb F_p$ such that $a^n\in\mathbb F_p$

Is it true that for every $n\in \mathbb N$ there exists a prime $p$ such that the extension $\mathbb F_{p^n}/\mathbb F_p$ has a primitive element $a\in \mathbb F_{p^n}$ and $a^n\in\mathbb F_p$? I ...
4
votes
2answers
52 views

Showing that the Field Extension $\mathbb{Q}(T^{1/4})/ \mathbb{Q}(T)$ is not Galois

Prove that $\mathbb{Q}(T^{1/4})$ is not Galois over $\mathbb{Q}(T)$, where $T$ is an indeterminate. I am not sure how to proceed due to the indeterminate. It suffices to show that the degree of ...
2
votes
0answers
53 views

A field F can be extended to one in which all polynomials over F have a solution

Please check to see if my methods are correct. This question is a lot to absorb all at once. Let P be the set of non-constant polynomials over a field F and let X be the set of finite subsets of ...
0
votes
1answer
39 views

Field extension generated by $\alpha$ and separability

In my notes, I have written that the field extension of $k$ generated by an element $\alpha \in K$, where $K$ a larger field, is defined to be $$k(\alpha) = \bigcap_{ \alpha \in E} E$$ where $E ...
5
votes
2answers
36 views

Galois group and degree of splitting field over complex rational functions.

Suppose $F=\mathbb C (t)$ the field of rational functions over $\mathbb C$. let \begin{equation*}f(x)=x^6-t^2\in F[x]\end{equation*}Denote $K$ as the spliting field of $f$ over $F$. I'm trying to ...
2
votes
2answers
50 views

The Galois group of a specific polynomial $ f(x) = x^6-2x^3+2 \in Q[x] $

Hello all I was given this question in Algebra asking me to show a polynomial's Galois group has both a subgroup and quotient group isomorphic to $S_3$ the three symmetric group. The polynomial in ...
0
votes
2answers
42 views

Example check: two algebraically closed fields with one a subset of the other

The problem asks to find two different algebraically closed fields $\mathcal{E}$ and $\mathcal{F}$ with $\mathcal{E} \subseteq \mathcal{F}$. We have not done a whole lot of stuff with algebraically ...
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2answers
39 views

What is the characteristic of a field in lay terms?

And why do we usually assume a field has characteristic not equal to 2?
3
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0answers
15 views

Uses for coefficients other than the trace and norm for an element belonging to a field.

The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the ...
3
votes
1answer
35 views

Question on finite fields and their extensions

I have been given this question in Algebra class on finite fields which I have tried to solve but to no avail, so all help appreciated. I am given $ p=13;q=p^6 $, then I am asked to prove or give a ...
3
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0answers
22 views

$O_S$ is the integral closure of $k[T]$ in $F$ for some embedding of $k(T)$ in $F$?

Let $F$ be a function field in one variable over a field $k$. Let $S$ be a nonempty finite subset of the set of all places of $F$. Let$$O_S = \{f \in F: \text{ord}_v(f) \ge 0 \text{ for all }X ...
2
votes
0answers
31 views

Calculating the Galois group of a covering map

Suppose $C$ is an algebraic curve and $\phi:C\rightarrow \mathbb{P}^{1}$ is a covering map of the complex projective line ramified at $\{0,1,\infty\}$ only. Suppose $\phi':C'\rightarrow ...
3
votes
1answer
45 views

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Problem: Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian. Prove that the Galois group of the field extension $K(\alpha + ...
1
vote
2answers
42 views

Let $K=GF(2)$ and $p(x)= x^3 + x+1.$ Show that $p$ is irreducible in $K[x]$

Let $K=GF(2)$ and $p(x)= x^3 + x+1$ Show that $p$ is irreducible in $K[x]$ First of all am I right in interpreting: $$GF(2) = \mathbb Z / 2 \mathbb Z= \{ 0,1\}$$ So basically, $p(x)$ is a ...
2
votes
1answer
61 views

All primes $p$ for which $x^{2} +1$ irreducible over $\mathbb{F}_{p}$

Let $\mathbb{F}_{p}$ be the field with $p$ elements. Determine all primes $p$ for which $x^{2}+1$ is irreducible over $\mathbb{F}_{p}$. Here's what I've got. I think this might be finished but I'm ...
1
vote
6answers
102 views

In a field $F=\{0,1,x\}$, how does $1 + 1 = x$?

I understand that in a field with two elements $1 + 1 = 0$, but in a field with three I do not understand how $1 + 1 =x$. The work I have done so far is: \begin{align} 1 + 1 &= \{ 0 , 1 , x\}\\ 1 ...
11
votes
1answer
227 views

Galois Group of $x^{4}+7$

I am trying to find the Galois group of $x^{4}+7$ over $\mathbb{Q}$ and all of the intermediate extensions between $\mathbb{Q}$ and the splitting field of this polynomial. I have found that the ...
6
votes
2answers
73 views

Embedding $\mathbb{F}_{q^2}^*$ into $GL_2(\mathbb{F}_{q})$

If we see $\mathbb{F}_{q^2}$ as a $2$-dimensional vector space over $\mathbb{F}_{q}$ (and pick a base) then we can identify $\operatorname{Aut}_{\mathbb{F}_{q}}(\mathbb{F}_{q^2})$ with ...
8
votes
2answers
72 views

Irreducibility of cyclotomic polynomials over number fields

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at ...
0
votes
1answer
26 views

Radical extension and algebraic solution of an irreducible polynomial

Suppose that $k$ is a field with characteristic equal to zero, that $P \in k[X]$ is an irreducible polynomial and that $\alpha$ is a root of $P$ in an algebraic closure $\overline{k}$. Suppose also ...
1
vote
1answer
38 views

Every irreducible polynomial f over perfect field F is separable

Every irreducible polynomial f over perfect field F is separable. Can you check my proof? Let f is inseparable. So we have $f=\sum_i h_ix^i$ and $f^p=\sum_i h_i^px^{ip}$ Now I use Frobenius mapping ...
1
vote
1answer
39 views

Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
1
vote
1answer
55 views

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal ...
2
votes
0answers
40 views

Calculus on Spaces other than $R$

I was thinking that all that the real numbers (and the complex numbers similarly) need for calculus to work is for them to be a field and a complete metric space, and probably have the usual field ...
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1answer
57 views

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$ My answer: Must show: i) $0 \in \mathbb{Q}{[\sqrt3]}$ ii) $1 \in ...
5
votes
1answer
58 views

Does it make sense to talk about complex matrices over the field of real numbers, R?

I don't see an issue with considering a vector space of complex matrices over R -- addition of matrices makes sense, but scalar multiplication will be done with real numbers. But I wanted to ask, ...
2
votes
1answer
43 views

extend trace and norm from a number field $K$ to $ K \otimes_{\mathbb Q} \mathbb R $

I read somewhere that we can extend the trace and the norm of a number field $K$ to the commutative algebra $ V=K \otimes_{\mathbb Q} \mathbb R$. Before state exactly my question, let me write the ...
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0answers
37 views

Cardinality of the set of all field automorphisms of $\mathbb C$ [duplicate]

Does $\mathbb C$ have infinitely many field automorphisms? Does it have uncountably many field automorphisms?
3
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0answers
22 views

For any field $k$ and $n>0$: $X^n-a\text{ reducible }\ \Leftrightarrow\ a=b^p\ \vee\ a=-4b^4$.

I'm stuck on the following exercise: Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such ...
2
votes
1answer
19 views

Unit group of a field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isnt $k = ...
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votes
0answers
17 views

Trace of an element in a separable field extension

Let $L=K(\alpha)$ be a finite separable field extension of $K$ of degree $n$ and let $\alpha$ have minimal polynomial $f(X)\in K[X]$ with roots $\alpha=\alpha_1,...,\alpha_n$. Write ...
2
votes
1answer
30 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
0
votes
1answer
16 views

connection between absolute irreducibility and smooth+geometrically connected

Let $C_1$ and $C_2$ be two smooth, projective, and geometrically connected curves over a field $K$ of characteristic $0$. Assume there are two non-constant branch mapping of $K$-curves, $\phi_1\colon ...
0
votes
1answer
44 views

Is there a way to generate groups, rings, fields, etc.? [closed]

There are ways to generate a list of numbers $a_1, a_2,...a,n$ such that no $a_i,a_j$ share any factors, mainly by letting $a_{n+1}=a_n^2-a_n+1$ with $a_0>1$, my question is: is there a way to ...
8
votes
1answer
81 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
0
votes
2answers
66 views

Create a field from set of 2 elements.

Can we always create a field from a set of at least $2$ elements? For addition I considered a function: $A\times A \rightarrow A$. If $a+b=b+a=a+a \rightarrow a $. If $b+b \rightarrow b$. Is it ...
1
vote
0answers
19 views

Computing number of irreducible polynomials of degree n over $\mathbb{F}_q$

When I try to find the number of irreducible polynomials (of degree n) over a finite field I first look for the number of $\alpha \in \mathbb{F}_{q^n}$ such that ...
0
votes
2answers
37 views

Showing that any field extension of a finite field is simple

We know that the multiplicative subgroup $F^\times$ of a finite field $F$ is cyclic. Use this to show that any field extension of a finite field is simple. Any clues?
2
votes
0answers
29 views

Intermediate “prime” extensions [Rotman]

Problem Assume $F$ contains the $k$th roots of unity, and let $R=F(\alpha)$, where $\alpha$ is a root of $x^k-a$ for some $a\in F$. Prove that there exist intermediate fields $$F=K_0\subset ...