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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-3
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1answer
47 views

Prove that $R$ is also a field. [closed]

Let $K$ be an algebraic extension of $F$ and let $R$ be a subring of $K$ and $F \subseteq R \subseteq K$. Then prove that $R$ is also a field.
0
votes
1answer
23 views

Prove that $E=F[\alpha^2]$ [duplicate]

Let $E=F[\alpha]$, $\alpha$ is algebraic over $F$ and $[E:F]$ is odd. Prove that $E=F[\alpha^2]$. Now clearly $[F[\alpha^2]:F]|[E:F]$ so $[F[\alpha^2]:F]$ is also odd. But how can we show that ...
4
votes
3answers
82 views

Show that $\mathbb{Q}(\sqrt{2},\sqrt{3},\dots,\sqrt{p},\dots)$ is an algebraic extension of $\mathbb{Q}$, for $p$ prime.

I have shown that $[\mathbb{Q}(\sqrt{2},\dots,\sqrt{p},\dots):\mathbb{Q}]=\infty$, by showing that $$ \mathbb{Q} \subset \mathbb{Q}(\sqrt{2})\subset \mathbb{Q}(\sqrt{2},\sqrt{3})\ldots$$ is an ...
1
vote
1answer
28 views

A question regarding proving the fact that every finite field is perfect

I am trying to prove the fact that every finite field is perfect. Hence, every irreducible polynomial is separable (does not have a repeated root). This is easy to prove when in a field of ...
1
vote
0answers
18 views

determinant of independent set of triangles

let $n > 3$. To a square free trinomial $x_i x_j x_k$ associate the $n$ vector that has all entries zeros except in the $i$-th, $j$-th, and $k$-th entries, where it has the value $1$ (i.e. all ...
11
votes
3answers
1k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
5
votes
2answers
44 views

Determining the minimal polynomial over $\Bbb{Q}$

I was working on a homework assignment from Hungerford: Find the minimal polynomial of the element $\sqrt{1+\sqrt{5}}$ over $\Bbb{Q}$. Naturally the solution would be the polynomial with roots ...
0
votes
1answer
16 views

Showing a set with operations comprise a non-ordered field

Given set $S = \{0,1\}$, and operations $+$ and $*$ given by the following identities: $$0+0=1+1=0;1+0=0+1=1;0*0=1*0=0*1=0;1*1=1.$$ I am trying to show $(S,+,*)$ is a field but not an ordered field. ...
3
votes
2answers
38 views

Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings?

Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings? If so, write down an explicit isomorphism. If not, prove they are not. My Try: Since $x^2+2$ is ...
2
votes
1answer
52 views

Why does this two-element field not have a supremum if we disregard the Completeness Axiom for the reals?

I have two questions, and I address them in two seperate paragraphs below. I read that if we do not accept the Completeness Axiom for the reals and fields but accept all the other axioms for fields ...
3
votes
1answer
42 views

galois group of cubic polynomial with 3 real roots--no discriminant

I know it's a duplicate, but the other one is a year old and got an answer using methods far beyond the typical first year abstract algebra. If I have an irreducible polynomial over $\mathbb{Q}$ with ...
1
vote
1answer
49 views

Irreducible polynomial $X^q-2$ [duplicate]

Prove that the polynomial $X^q-2$ is irreducible in the ring $\Bbb Q(\sqrt[p]{2})[X]$ What method i can use for proving, that this polynomial is irreducible in this specific ring?
2
votes
1answer
19 views

E finite galois extension over Q of order pq^m, show irreducible f that splits in E is solvable by radicals

Let $E$ be a finite Galois extension of $\mathbb{Q}$ of degree $pq^m$, where p and q are prime such that $p<q$. I need to prove that every irreducible polynomial over $\mathbb{Q}$ that splits in ...
0
votes
1answer
35 views

Showing irreducibility of a polynomial. [duplicate]

How would you go about showing that $p(x)=\frac{x^5-1}{x-1}=x^4+x^3+x^2+x+1$ is irreducible over $\mathbb{Q}$. I'm having trouble seeing how one can show whether this kind of polynomials are ...
0
votes
0answers
23 views

Trace and Determinant of Field Extension

In algebra, we had a look at the trace and the determinant of a field extension. I am familiar with those concepts in linear algebra and I have seen that finite extensions can be viewed as a finite ...
1
vote
1answer
33 views

We can code the integers into the orders at zero of elements of $F_q(t)$

There is the following part in the paper that I am reading: We can code the integers into the orders at zero of elements of $F_q(t)$ (the field of rational functions in $t$ with coefficients in a ...
2
votes
2answers
75 views

Show that $[\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3}):\mathbb{Q}(\sqrt[3]{2})] \gt1$ [closed]

I have to show that the degree $[\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3}):\mathbb{Q}(\sqrt[3]{2})]$ is $\gt1$. I know that for this purpose it is enought to show that $\sqrt[3]{3} \notin ...
0
votes
1answer
38 views

How would I find the Field's elments generator?

Suppose we would construct a Field $F=GF(2^4)$ by using $f(x)=x^4+x^3+x^2+x+1$. In this case the generator is $\alpha=x+1$. Why $\alpha$ here is equal to $x+1$ how would I find this?
2
votes
1answer
35 views

Proof Check: Number of elements of $\mathbb{F}_{p^{n}}$ of the form $a^{p}-a$ for some $a \in \mathbb{F}_{p^{n}}$.

Consider the map $\varphi:\mathbb{F}_{p^{n}} \rightarrow \mathbb{F}_{p^{n}}$ defined by $x \mapsto x^{p}-x$. Since $(a+b)^{p}= a^{p}+b^{p}$ for all $a,b \in \mathbb{F}_{p^{n}}$ we have that $\varphi$ ...
0
votes
1answer
17 views

Frobenius tower terminates to the identity after finite number of steps

Let $k$ be a field of characteristic $p>0$ and let $\phi: k \to k$ be the frobenius homomorphism $a \mapsto a^p$. I want to show that for every element $a \in k$ the sequence $\phi^n (a)$ ...
0
votes
0answers
36 views

Irreducible polynomial over a perfect field of characteristic $p\neq 0$

Let $E/F$ be an algebraic field extension where $F$ is of characteristic $p\neq 0$. Let $F'=\{x\in E : x^{p^n}\in F \;\text{for some}\;n\geq 0\}$. Then $F'$ is a perfect field containing $F$ and $E$ ...
1
vote
2answers
39 views

Finding dimension of a field extension

How would anyone go about this problem? Find dim$_\mathbb{Q}\mathbb{Q}(\alpha,\beta)$ where $\alpha^{3}=2$ and $\beta^{2}=2$. Thanks for your help, I really don't know how to go about this ...
0
votes
0answers
26 views

Arithmetic data in an elliptic curve over a field $\mathbb K$

Note: In this context, $E(K)$ denotes an elliptic curve $E$ over a number field $K$, and $L(E,s)$ denotes the Hasse-Weil $L$ function. Is the rank of the abelian group $E(K)$ of points of $E$ the ...
0
votes
1answer
61 views

Degree on Galois theory

Let $p$,$q$ prime numbers.How i can estimate the degree $[\Bbb Q(\sqrt[p]{2},\sqrt[q]{2}):\Bbb Q(\sqrt[p]{2})]$.Can anyone help me with giving me any hints for finding this degree?
0
votes
0answers
26 views

Smoothness in cyclotomic versus complex fields?

Say we have a polynomial in a cyclotomic field; in particular, an n-th cyclotomic field, where n is the order of the polynomial's symmetry group. If we know the polynomial is smooth over this field, ...
3
votes
2answers
41 views

Maximal unramified extension of a global function field

Can we explicitly describe the unramified extensions of a global function field, for instance $\mathbb{F}_q(T)$?
3
votes
0answers
24 views

Show that every algebraic field extension has a normal closure

I am trying to show that every algebraic extension of fields has a normal closure. Let $L/K$ be an algebraic extension. First suppose that the extension is finite. So $L=K(\alpha_1,...,\alpha_n)$ ...
6
votes
3answers
3k views

Subfields of finite fields

We know that if a finite field $F$ has characteristic $p$ (prime), then $F$ has cardinality $p^r$ where $r = [F:\mathbb{F}_p]$. I'm now trying to say something about the possible cardinalities of ...
2
votes
1answer
29 views

Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
1
vote
1answer
34 views

Extending a Field Monomorphism

In theorem A3.5 of Ash's book Abstract Algebra: The Basic Graduate Year (page 20 in this pdf), the author set out to prove the following. Let $\sigma: F \rightarrow L$ be a field monomorphism ...
3
votes
1answer
36 views

Is $K := \mathbb{Q}(\cos (2\pi / 11))$ a Galois extension over $\mathbb{Q}$?

I believe that it is because $\cos(2\pi / 11) = (\zeta + \zeta^{-1})/2$ where $\zeta = e^{2\pi i/11}$ is a primitive $11$-th root of unity, and so $K$ is a subfield of $\mathbb{Q}(\zeta)$ with ...
2
votes
1answer
50 views

Associate a discrete valuation ring to a field $k$.

I have a field $k$ of positive characteristic $p$, not necessarily perfect. Can i find a discrete valuation ring that have $k$ as residue field and field of fractions $K$ of characteristic zero? I ...
1
vote
1answer
26 views

Galois group of maximally tamily ramified extension over the maximally unramified extension of a global function field F

Suppose $F$ were a local nonarchimedean field of characteristic zero of residue characteristic $p$. Let $F^{un}$ the maximally unramified extension. Let $F^{un}\subset F^{tame}$ be the maximally tame ...
6
votes
1answer
57 views

Existence of Jordan decomposition over finite field

Prove that over finite field $\mathbb F$ exists additive Jordan-Chevalley decomposition: for all matrix $M$ there are semisimple matrix $M_{s}$ and nilpotent matrix $M_{n}$ such that $M=M_{s}+M_{n}$. ...
2
votes
0answers
25 views

Question about the index of a subgroup in $\mathrm{Aut}(\mathbb{C} / K )$ with $K$ a number field.

Suppose that $k_0$ is a number field with subfield $K$. Set $[k_0 : K] = d$. If $G = \mathrm{Aut}(\mathbb{C} / K )$ and $H$ is the subgroup of $G$ which fixes $k_0$, is it true that $[G:H] = d$? ...
1
vote
1answer
71 views

Factoring $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
2
votes
1answer
47 views

Problem involving cubic field extensions

Let $F$ be a field of characteristic $0$ and let $L$ be a cubic extension. I want to show that there exists an element $a \in F,$ and an extension $L_0$ of $\mathbb{Q}(a)$ such that ...
3
votes
1answer
148 views

Can extension by an isomorphic field be of degree at least 2?

Suppose $K/F$ is a field extension such that $K\not=F$. Is it legitimate to say that $F$ and $K$ can't be isomorphic since by assumption \begin{equation*}[K:F]\ge 2\end{equation*}and if $K$ and $F$ ...
3
votes
3answers
69 views

Why does $\sqrt{6} + \sqrt{10} + \sqrt{15}$ have four conjugates?

I am having trouble understanding how algebraic number $\sqrt{6} + \sqrt{10} + \sqrt{15}$ has four conjugates. Minimal polynomial is $x^4-62 x^2-240 x-239$ according to Wolfram Alpha. Factorized: ...
3
votes
1answer
47 views

What do we know about fields possessing an involution?

The field $\mathbb{C}$ of complex numbers has an involution, and the same is true of the field of algebraic numbers (the algebraic closure of $\mathbb{Q}$ as a subfield of $\mathbb{C}$) and of the ...
6
votes
2answers
110 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
1
vote
1answer
39 views

Is there an isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$ for primes $p \neq q$?

Let $p \neq q$ be distinct primes. Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$? Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}$? If such an isomorphism exists, given ...
42
votes
6answers
4k views

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
2
votes
0answers
42 views

Proving that $t^{p^r}-a$ is irreducible when $a\in k$ is not a $p$th power

Let $p$ be an odd prime, $F$ a field of characteristic $0$ and $a\in F$ with $a\neq 0$. Assume $a$ is not a $p$th power in $F$. Prove that for every positive integer $r$, $t^{p^r}-a$ is irreducible ...
6
votes
1answer
83 views

Is there an infinite topological meadow with non-trivial topology?

For reference meadows are a generalization of fields that were designed to be compatible with the requirements of universal algebra. Specifically a meadow is a commutative ring equiped with an ...
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vote
0answers
60 views

Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
2
votes
1answer
28 views

Basis for the field extension $\mathbb{Q}(\zeta_{12})$

Consider the cyclotomic field $\mathbb{Q}(\zeta_{12})$ where $\zeta_{12}$ represents the $12$-th primitive root of unity. Since the minimal polynomial of $\zeta_{12}$ is given by $\Phi_{12}(x)$ which ...
1
vote
0answers
34 views

$x^p -x-1$ irreducible over $\mathbb{F}_{p}$ [duplicate]

Show that $x^p - x -1$ is irreducible over $\mathbb{F}_{p}$. I've seen this polynomial (or some variation x^p -x -a) on several of our qualifying exams and in every case they ask you to show it is ...
2
votes
3answers
234 views

Can we say “commutative ring = field”?

We know the difference between ring ($R$) and field ($F$) is that $R$ does not guarantee multiplication is commutative. Now, if considering commutative $R$, which means ($R$, $*$) is a group, can ...
1
vote
1answer
27 views

A question about fields and separability in Serre's “Local Fields”

On page 14 of the English edition of Serre's "Local Fields", that is chapter 1, section 4, I am confused by the following; there is talk of fields $B/\mathfrak P$ and $A/\mathfrak p$ for prime ideals ...