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 ...
5
votes
4answers
193 views
Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.
Show $p(x) = x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.
By Gauss' Lemma, $p(x)$ is irreducible in $\mathbb Q[x]$ if and only if it is irreducible in $\mathbb Z[x]$. We can look at ...
3
votes
1answer
65 views
Subfield of rational function fields
Let $k\subset F\subseteq k(x_{1},x_{2},...,x_{n})$ where $k$ and $F$ are the fields and $x_{1},x_{2},...,x_{n}$ are transcendental over $k$. Can we express $F$ in terms of or function of ...
1
vote
4answers
80 views
Let $\mathbb{F}$ be any field. Show that the number of cube-roots of unity in $\mathbb{F}$ is either $1$ or $3$.
Let $\mathbb{F}$ be any field.
Show that the number of cube-roots of unity in $\Bbb F$ is either $1$ or $3$.
Show that if $\mathbb{F}$ has characteristic $3$ then it has only one cube-root of ...
1
vote
2answers
43 views
Struggling with a question on quotients in elementary Galois theory
I have started teaching myself Galois Theory. I have a problem understanding a part of the proof of the following proposition : Let $K\subseteq L$ be a field extension and $l\in L$ an element which is ...
1
vote
2answers
59 views
Subfields of Rings
I am currently working through an undergraduate class in Galois Theory. I have come across a question that I am unsure about.
Can a ring that is not a field, have a subring that satisfies the ...
4
votes
2answers
66 views
Transcendental extension of field.
Let $F(a)$ be a transcendental extension of the field $F$. Given an element $b \in F(a)$ such that $b \notin F$, I would like to show that $F(a)$ is an algebraic extension of $F(b)$.
My idea of ...
2
votes
2answers
81 views
If $\mathrm{char}(K)=p$ is prime, $L/K$ is separable if and only if $K(\alpha) =K(\alpha^p)$ for all $\alpha \in L$ [duplicate]
I am trying to prove that if $L/K$ is an algebraic extension and if $\alpha \in L$, then
$\alpha$ is separable over $K$ if $\mathrm{char}(K)=0$. This is clear because $K$ is perfect which in turn ...
3
votes
1answer
59 views
When are powers of primitive elements still primitive elements
This question is motivated by this question and is tangentially related to this question.
Let $L/K$ be a finite Galois extension of fields. Pick $\alpha \in L \setminus K$ and consider the simple ...
2
votes
1answer
102 views
Problem with roots of unity
Let $\zeta$ a root of $x^{p}-1$, with $p$ an odd prime, and $K$ a subgroup of the mutiplicative group $\mathbb{Z}_p^{*}$ of index $2$. I need to prove that
$a=\displaystyle\sum_{k\in K}\zeta^{k}$
...
1
vote
2answers
59 views
Transcendental elements over field extensions.
Let $E/F$ be a field extension, and suppose $a \in E$ is transcendental over $F$. I'm reading a proof that says $\dim_F F(a) \ge \dim_FF[x] = + \infty$ since the evaluation map $F[x] \to F [a]$, ...
2
votes
1answer
59 views
Injection from an integral domain to its field of fractions.
I have a quick question about modules. Suppose that $R$ is an integral domain with field of fractions $K$. Then any free $R$-module is isomorphic to copies of direct sums of $R$, say $R^i$ . ...
2
votes
1answer
75 views
Countable Field Extension of a Countable Field
Okay, first question on this site, apologies in advance for any mistakes I may make.
Question:
So I need to show that an algebraic field extension $E:F$, with $F$ being countable, is countable.
My ...
2
votes
3answers
132 views
Field extensions that are not normal
I am trying to come up with field extensions $M : L : K$ such that none of the three extensions $M:L, L:K, M:K$ are normal.
So far, I have tried letting $K = \mathbb{Q}, L = \mathbb{Q}(\sqrt[3]{2})$. ...
7
votes
2answers
121 views
Explicit Galois theory computation in cyclotomic field
Let $p$ be an odd prime. Let $\zeta=e^{\frac{\pi i}{4 p}}$, thus $\zeta$ is a primitive $8p$-th root of unity. There is a unique $\mathbb Q$-automorphism $\tau$ of the number field ${\mathbb ...
1
vote
4answers
102 views
Polynomial factorization over finite fields
How can i factorize the polynomial $x^{12}-1$ as product of irreducibles polynomials over $\mathbb{F}_4$? Anyone can help me?
4
votes
1answer
165 views
Irreducible cyclotomic polynomial
I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
1
vote
2answers
88 views
Galois group of a polynomial
I want to know how to find a polynomial $f(x)$ of degree $5$ in $\mathbb{Q}[x]$ with Galois groups $G_f=\mathbb{S}_5$
21
votes
6answers
534 views
Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?
It seems to me (intuitively) that there should be no other fields whose algebraic closure is $\mathbf{C}$, even though I have no reason to believe it. The facts I've been using to formulate an ...
5
votes
3answers
79 views
Lifting isomorphisms of fields to automorphisms of polynomial rings
Let $L$ be a field and $\alpha, \beta$ algebraic over $L$ such that $L(\alpha)\cong L(\beta)$. If $q(t)$ and $p(t)$ are the minimum polynomials of $\alpha$ and $\beta$, respectively, does it follow ...
2
votes
1answer
97 views
Subfield of finitely generated field over $k$
Let $k$ be a field. Let $k(x_{1}, x_{2},...,x_{n})$ be a finitely generated field over $k$. Where $x_{1}, x_{2},...,x_{n}$ are transcendental over $k$. Let $F$ is a field which is in between $k ...
1
vote
1answer
64 views
Are all functions on vectors in GF(2^n) representable as polynomial functions?
In $GF(2)$, any function from an n-dimensional vector to a number is equal to a polynomial function of n variables. (See proof below.)
Question: Is this true for other $GF$, especially $GF(2^8)$? ...
1
vote
1answer
59 views
Splitting field of a polynomial in an extension of degree 2 in characteristic 2
Let $K \subset L$ be an extension of degree 2. If $\operatorname{char}(K)=2$ then there exists $a \in K$ such that $L$ is the splitting field over $K$ of a polynomial of the form $X^2-a$ or $X^2-X-a$.
...
5
votes
2answers
209 views
Determining subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ without Galois theory?
Somehow I had convinced myself recently that one could determine the subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (here $\zeta_3 = \frac{-1+\sqrt{3}i}{2}$ is a primtive $3$rd root of $1$) without ...
4
votes
2answers
75 views
Is there a field extension over the real numbers that is not the same as the field of complex numbers?
I'm trying to determine if there is a field, F, such that $\mathbb{R}$ $\subsetneq$ F $\subsetneq$ $\mathbb{C}$ where F is not the same as $\mathbb{R}$ or $\mathbb{C}$.
1
vote
2answers
67 views
Question about algebraic closures
Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $K = \mathbb{Q}(\sqrt{d})$ and $\overline{K}$ defined to be the algebraic closure of $K$. Is it true that $\overline{K} \cong ...
1
vote
1answer
123 views
Finite Subgroups of Multiplicative Group of Field
Question:
Let F be a field of characteristic $0$ such that $|F:\mathbb Q|=2$, and let U be a finite subgroup of F*, the multiplicative group of F. Show that $|U|$ is 1, 2, 3, 4 or 6.
Attempt at ...
1
vote
2answers
65 views
Embedding ordered number fields into $\Bbb R$
Consider a number field $K$ equipped with a total order $\le$. Is there always a field homomorphism $\phi:K \rightarrow \Bbb R$ which respects the total order, i. e. with $\phi(x)>\phi(y)$ for ...
1
vote
2answers
68 views
Normal field extension separable over its fixed field
Let us have a field $K\supseteq E$ and $G$ be its group of automorphisms over $E$. Let the fixed field of $G$ be $K^G$. I would like to show that $K$ is separable over $K^G$.
I know that for ...
4
votes
0answers
123 views
Proof that a finite separable extension has only finite many intermediate fields
Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
0
votes
2answers
49 views
Two different definitions of separable polynomial
This is from A field guide to Algebra by Antoine Chambert Loir.
A polynomial $P \in K[X]$ is separable if and only if its roots are in an algebraic closure of $K$ are simple.
Here is ...
0
votes
1answer
23 views
Showing equality of a certain ring and a field
Let $K \subset E $ be a field extension and $S \subset E$ then define $K[S]$ and $K(S)$ as the smallest subring and subfield of E respectively that contains K and S. I want to show the following ...
2
votes
2answers
127 views
Field of characteristic 0 such that every finite extension is cyclic
I am trying to construct a field $F$ of characteristic 0 such that every finite extension of $F$ is cyclic. I think that I have an idea as to what $F$ should be, but I am not sure how to complete the ...
0
votes
1answer
147 views
Show that a number field is isomorphic to a quotient $\mathbb Q[x]/(f)$
Let $K$ be a number field of degree 3. Show that $K$ is isomorphic to
a quotient $\mathbb Q[x]/(f)$, with $f = x^3 + ax + b$ in $\mathbb Z[x]$ irreducible in $\mathbb Q[x]$ (without using the result ...
0
votes
2answers
61 views
Splitting of polynomial into linear factors
I am trying to show that the polynomial $x^3 - 3$ splits into linear factors over $ \mathbb{Z}_7[x]/ \langle x^3 - 3 \rangle $, but am having trouble doing so, as I'm not very familiar with rings.
...
3
votes
1answer
51 views
Finding degree of the extension [duplicate]
Is it true that the degree of extension $\mathbb Q(\sqrt {2},\sqrt {3},\sqrt {5},\dotsc,\sqrt {p_n}) / \mathbb Q$ is $2^n$ where $p_n$ is the $n$th prime number. If so, how to prove this? My idea is ...
10
votes
1answer
81 views
Other Euler characteristics?
At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
0
votes
2answers
85 views
basis and dimension of the splitting field of $x^4+5x^2+6$
Please help me finding the basis and dimension of the splitting field of the polynomial $x^4+5x^2+6$ in the rational field.
Thanks
1
vote
2answers
70 views
Why this polynomial is irreducible?
Let $K=\mathbb{Z}_p(t)$, how to prove $f(x)=x^p-t$ is irreducible in $K[x]$?
1
vote
1answer
45 views
Norm -1 in the extension $\,E[i]/E\,$ , where $\,E=\Bbb Q(\zeta)\,,\,\, \zeta^5 = 1$
Denote by $\zeta = \exp(2\pi i/5)$ the primitive root of unit of order 5 ($\zeta^5=1, \zeta \ne 1$). Let $E = \mathbb{Q}[\zeta]$. Then $i = \sqrt{-1} \notin E$. Let $L = E[i]$. We want to show that ...
3
votes
4answers
129 views
Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing
I need to rationalize $\displaystyle\frac{1}{a+b\sqrt[3]2 + c(\sqrt[3]2)^2}$
I'm given what I need to rationalize it, namely ...
6
votes
2answers
76 views
Equivalent definition of algebraically closed
In Hungerford's Algebra text, it is stated that a field $K$ is algebraically closed iff
there exists a subfield $F$ such that $K$ is algebraic over $F$ and all polynomials in $F[x]$ split in $K[x]$. ...
4
votes
5answers
93 views
Field extension of composite degree has a non-trivial sub-extension
Let $E/F$ be an extension of fields with $[E:F]$ composite (not prime). Must there be a field $L$ contained between $E$ and $F$ which is not equal to either $E$ or $F$? To prove this is true, it ...
2
votes
2answers
53 views
No roots over $F_2[X]/(X^3+X+1)$ [duplicate]
Possible Duplicate:
Reducibility over a certain field.
I am new to field theory. How can I show that $X^4+X^2+1$ has no roots in $F_2[X]/(X^3+X+1)$? All I know at this moment is that it is ...
2
votes
2answers
69 views
Roots of a non-separable irreducible polynomial
If we are working in a field of characteristic $p>0$, $K$, and we have $f$ irreducible and non-separable over $K$. If we have $L/K$ be the splitting field of $f$ and $r$ a root of $f$, is it true ...
3
votes
2answers
192 views
Dimension of a splitting field of a cubic polynomial over $\mathbb{Q}$
I know that for the cubic polynomial $x^3-5$ over $\mathbb{Q}$ the splitting field is $\mathbb{Q}(\sqrt[3]{5},e^{\frac{2\pi i}{3}})$, but I cannot convince myself why the dimension of this splitting ...
2
votes
3answers
137 views
Is any field contained in $\mathbb{C}$ up to isomorphism?
To be more specific, given a field $k$, must we have the algebraic closure $\bar{k}$ contained in $\mathbb{C}$ up to isomorphism?
2
votes
1answer
29 views
Possibilities for $[K^{sep}:K]$
I use the notation $K^{sep}$ to denote the separable closure of a field $K$ and $\bar{K}$ for the algebraic closure, that is
$$K^{sep}:=\{\alpha \in \bar{K} \mid \alpha \text{ is separable over } ...
1
vote
1answer
43 views
Reducibility over a certain field.
Let $K=F_2[x]/(x^3+x+1)$. I want to show that $f(x)=x^4+x^2+1$ is reducible over $K$ but has no roots in it. How to proceed? I know that $F$ contains 8 elements, how is the structure of these ...
14
votes
3answers
262 views
Finding the degree of a field extension over the rationals
Let $\alpha = \sqrt{\sqrt{2}+\root 4 \of 2}$, $\beta =\sqrt{\sqrt{2}- \root 4 \of 2}$, $\gamma = \sqrt{-\sqrt{2}+i\root 4 \of 2}$ and $\delta = \overline{\gamma}=\sqrt{-\sqrt{2}-i\root 4 \of 2}$.
Let ...
4
votes
1answer
65 views
An element is integral iff its minimal polynomial has integral coefficients.
This is from Algebraic Number Theory by Neukirch
Let $A$ be an integral
domain which is integrally closed, K its field of fractions, $L|K$ a finite
field extension, and $B$ the ...
