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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8
votes
5answers
477 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 ...
2
votes
2answers
110 views

No rational solutions of a system of equations

Please show that there does not exist $(a,b,c)\in\mathbb{Q}^3$ such that \begin{matrix} a^2b+2b^2c+2ac^2=0\\ a^2c+ab^2+2bc^2=0\\ a^3+2b^3+4c^3+12abc=3. \end{matrix} I'm able to show that this ...
1
vote
0answers
96 views

Showing $f(x)=g(x^{p^a})$ over field of (prime) characteristic $p>0$.

Let $f$ be a non-constant irreducible polynomial over a field $F$ of (prime) characteristic $p>0$. I need to prove that $f$ can be presented as: $f(x)=g(x^{p^a})$, where $g$ is irreducible over ...
0
votes
0answers
81 views

Quadratic equation proof in field with characteristic field $\neq 2$

Suppose that I have a field $F$ with $char(F)\neq 2$. How one can prove quadratic formula in that field(?)? Thank you.
10
votes
3answers
888 views

Does every algebraically closed field contain the field of complex numbers?

Does every algebraically closed field contain the field of complex numbers? Thank you very much.
7
votes
1answer
564 views

An exercise on cyclic extensions of Hungerford's book, Algebra.

I'm trying to show the following exercise of the book, it is in the section of cyclic extensions and says the following: Let $\overline{\mathbb Q}$ be a fixed algebraic closure of $\mathbb Q$, let ...
1
vote
2answers
643 views

Roots of polynomials over finite fields

I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$. I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
0
votes
1answer
54 views

Polynomials decomposition into irreduceables

I've been trying to find the composition to irreduceables of the following polynomials with no much success: X^2 +1 over the field F7 and X^2-2 over the field F5 Is there any method/algorithms I ...
2
votes
1answer
206 views

Field extension of $\mathbb Q$ of degree 2

Assume $K:\mathbb Q$ is a field extension and $[K:\mathbb Q] = 2$. Show that there is a unique squarefree $d \in \mathbb Z$ such that $K = \mathbb Q(\sqrt d)$. I know that $K$ is generated by say ...
2
votes
1answer
144 views

The number of subfield $K$ of $L$ such that $\,\mathbb{Q}\subsetneq K\subsetneq L$

$\omega\neq 1 \in\mathbb{C}$ such that $\omega^3=1$, suppose $L$ be the field generated by $\omega, 2^{1/3}$ over $\mathbb{Q}$ i.e $L=\mathbb{Q}, (\omega,2^{1/3})$, the number of subfields $K$ of $L$ ...
1
vote
1answer
79 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
2
votes
2answers
73 views

Special elements of fields extensions

I was wondering if there is a method to find all elements $w\in F(\alpha_1,\ldots,\alpha_n)$ such that $F(w)=F(\alpha_1,\ldots,\alpha_n)$, where $\alpha_1,\ldots,\alpha_n$ are algebraic over the field ...
8
votes
3answers
1k views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
2
votes
1answer
228 views

Characterization of transcendental elements in algebraic function fields

I would like to prove this equivalence: Let $F|K$ be an algebraic function field. Then $z \in F$ is transcendental over $K$ if and only if $[F:K(z)] < \infty$. (This statement is Remark 1.1.2 ...
8
votes
1answer
104 views

Help understanding fields.

Hi guys I have a test this tuesday and I am given practice questions to do , and I have trouble understanding fields. Like I know by definition what they are, but applying them is kind of confusing. ...
1
vote
2answers
73 views

Maps compatible with the Frobenius

Let $F$ be a field. Fix a separable closure $F^{sep}$. Consider the monoid whose elements are maps of sets $F^{sep} \to F^{sep}$ which are equivariant with respect to the Galois action. These maps ...
3
votes
0answers
129 views

Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
2
votes
1answer
399 views

Splitting field that isn't a Galois extension

I'm trying to find a counter-example to following statement: if $K$ is the splitting field of $g\in F[x],$ then the extension $K/F$ is Galois. I know the statement is true if $g$ is separable, ...
0
votes
1answer
64 views

Why does the minimal polynomial not change on changing the field?

Suppose $V$ is a vector space over a field $K$, $F$ is a subfield of $K$ and $T:V(F)\to V(F)$ is a linear operator. I think that if the field $F$ is changed to $K$ the characteristic polynomial of $T$ ...
1
vote
2answers
97 views

Is $x^2+1$ irreducible over a cyclotomic field?

Let $K=\mathbb{Q}[\omega]$, where $1+\omega+\omega^2=0$, let $f(X)=X^2+1$. How can i prove irreducibility of $f$ over $K$?
2
votes
2answers
139 views

Help Determining Degree of a Field Extension

Question: Determine the degree of $\mathbb{Q}(\alpha)$ over $\mathbb{Q}$, where $\alpha^3=2$. Determine the degree of the splitting field of $f(t) = t^3 - 2$ over $\mathbb{Q}$. Is there a difference ...
4
votes
0answers
556 views

Minimal polynomial of a finite purely inseparable field extension

Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
7
votes
0answers
216 views

Limits and colimits in the category of fields

It is said that the category of fields $\mathsf{Fld}$ is ill-behaved, for example it is not an algebraic theory, does not have initial or terminal objects. In particular it is not presentable. On the ...
1
vote
0answers
33 views

Let $F$ be a field and $E=F(a)$, where the min poly of a has degree n over F. What are the conditions on $m\in \mathbb{Z}$ so that $E=F(a)=F(a^m)$

So far I've been playing around with different techniques. I tried to show double inclusion to get the equality in hopes of coming across a needed condition on m, but was unsuccessull. I also worked ...
2
votes
1answer
328 views

Finitely generated integral domain and finitely generated $k$-algebra.

Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My ...
4
votes
1answer
56 views

No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$

Let $f(x) \in \mathbb Q[x]$ be irreducible of degree 4. Let $\alpha$ be a root of $f(x)$. Let $E = \mathbb Q(\alpha)$ and let $K$ be the splitting field of $f(x)$ over $\mathbb Q$. Prove that there is ...
1
vote
1answer
384 views

Degree of the splitting field of $x^{p^2} -2$ over $\mathbb{Q}$, for prime p.

I've already shown that the degree of the splitting field of $x^p-2$ over $\mathbb{Q}$ is $p(p-1)$ as follows: $x^p-2$ has roots $\sqrt[p]{2}\omega_{k}$ for $k=0,1,...,p-1$, where the $\omega_{k}$ ...
10
votes
3answers
302 views

Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
2
votes
2answers
664 views

Proving that the inverse of an algebraic element is algebraic

Let $E$ be an extension of a field $F$. Suppose $c \in E$ is algebraic over $F$ where $c \neq 0$. I want to prove that $c^{-1}$ is also algebraic over $F$. I feel like I'm missing something obvious ...
1
vote
0answers
226 views

Trace and cyclotomic field

Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that ...
7
votes
1answer
243 views

Show the two fields are not isomorphic

Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$. ...
3
votes
2answers
151 views

Why is $X^4 + \overline{2}$ irreducible in $\mathbb{F}_{125}[X]$?

I want to prove that $f = X^4 + \overline{2}$ is irreducible in $\mathbb{F}_{125}[X]$. I know that $\mathbb{F}_{125}$ is the splitting field of $X^{125} - X$ over $\mathbb{F}_5$, and that this is a ...
2
votes
2answers
161 views

Roots of unity in $\mathbb{Q}(\zeta_p)$ for $p$ an odd prime

I am confused about a last step in a proof that the only roots of unity in $\mathbb{Q}(\zeta_p)$ are $\pm\zeta_p^j$, where $\zeta_p=e^{2 \pi i/p}$, $p$ is an odd prime and $1\leq j\leq p-1$. So far ...
4
votes
0answers
216 views

Determining whether or not an extension is a splitting field

On an exam for my modern algebra class, we were asked to determine whether or not $\Bbb{Q}\left(\alpha\right)$ is the splitting field (here using the convention that splitting field means the smallest ...
0
votes
1answer
52 views

About extension of fields

Is there a field extension $L/K$ such that it is an infinite algebraic extension of fields but the separable degree of $L$ over $K$ is finite?
5
votes
6answers
396 views

Show that $x^4 +1$ is reducible over $\mathbb{Z}_{11}[x]$ and splits over $\mathbb{Z}_{17}[x]$.

Reduction into linear factors $\mathbb{Z}_{17}[x]$: This part is not too hard: $x^4 \equiv -1$ mod 17 has solutions: 2, 8, 9, 15 so $(x-2)(x-8)(x-9)(x-15) = x^4 -34 x^3 +391 x^2-1734 x+2160 \equiv ...
5
votes
1answer
99 views

A question on morphisms of fields

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields? My guess is yes, because every morphism ...
4
votes
1answer
344 views

Show that $x^4-10x^2+1$ is reducible over $\mathbb{Q}(\sqrt{6}), \mathbb{Q}(\sqrt{2}),$ and $\mathbb{Q}(\sqrt{3})$. Is there an easier method?

Ok, I actually worked this out as I was typing it up. But my solution seems kind of inelegant and involves a lot of tedious algebra that I've omitted here. Can anyone think of an easier method? (1) ...
2
votes
3answers
101 views

Show that for $n \geq 2$, the $n^{th}$ cyclotomic polynomial is a reciprocal polynomial, i.e. $\Phi_{n}(x) = x^{\phi(n)}\Phi(n)(x^{-1})$.

Here $\phi(n)$ is the Euler totient function and the degree of $\Phi_{n}(x)$. What I've done so far: Let $\phi(n) = p$ so the following products each have p components. $$\Phi_{n}(x) = \prod_{k=1, ...
0
votes
1answer
61 views

$KL$ is normal over $F$ if both $K$ and $L$ are normal over $F$

Let $K$ and $L$ be extensions of $F$. Show that $KL$ is normal over $F$ if both $K$ and $L$ are normal over $F$.
0
votes
1answer
297 views

A radical extension with a non-radical subextension

For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely ...
2
votes
1answer
177 views

Show the extension is not cyclic

Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
1
vote
2answers
222 views

Number of irreducible polynomials of degree $3$ over $\mathbb{F}_3$ and $\mathbb{F}_5$.

I'm trying to find the number of $3$rd degree irreducible polynomials over $\mathbb{F}_3$ and $\mathbb{F}_5$. Since a $3$rd degree polynomial is irreducible if and only if it is divisible by a ...
2
votes
3answers
90 views

Two quadratic fields over $\mathbb{Q}$

I'm having a bit of trouble showing that the two quadratic fields $\mathbb{Q}[X]/(X^2+1)$ and $\mathbb{Q}[X]/(X^2+3)$ over $\mathbb{Q}$ are not isomorphic (as fields). Could someone help me? Perhaps ...
1
vote
2answers
79 views

Convert polynomials and fractions in a finit field?

I am trying to understand how finite field works, and I am stuck on converting high power polynomials into a power of the field, also converting fractions into integers. $8^{-1}\cdot44$ in $\Bbb ...
0
votes
0answers
115 views

How to list all irreducible polynomials in a field?

I am currently trying to refresh my memory on some basic primary polynomials, apologies if my terminologies aren't correct: For example, I have a field $\Bbb F_{2^3}$ and generates a list of ...
4
votes
0answers
154 views

Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
2
votes
1answer
228 views

Fixed field of automorphisms determined by $t\mapsto at+b$.

Suppose $E=\mathbb{F}_p(t)$, the field of rational functions in a transcendental $t$ over the finite field of $p$ elements. Suppose $G$ is the group of field automorphisms fixing $\mathbb{F}_p$ ...
6
votes
1answer
261 views

Irreducibility over $\mathbb F_p$ - A useless hint?

Dummit and Foote, 13.5.5: For any prime $p$ and nonzero $a \in \mathbb F_p$ prove that $x^p-x+a$ is irreducible and separable over $\mathbb F_p$ The question goes on to suggest two approaches ...
4
votes
2answers
74 views

Is there a natural topological structure for a general field $\mathbb{K}$?

If not, what are some standard structures that define a topology?