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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4
votes
1answer
135 views

Help with a bilinear form

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m is odd) I need to prove that ...
5
votes
2answers
126 views

Trace function equation

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m odd), with $a,b\neq 0$. I need to prove that $$\sum_{i=1}^{(m-1)/2}\operatorname{tr}(a^{2^{i}}b+b^{2^{i}}a)=0\qquad \text{ iff }\qquad ...
5
votes
1answer
491 views

Calculating Splitting Field Degree of Extension

Is there an easier way to calculate the degree of extension of a splitting field for a polynomial like $$x^7-3\quad\text{over}\quad\Bbb Z_5\,?$$My approach for several of these have been to find all ...
3
votes
2answers
95 views

Why does each subfield have to contain multiples of 1?

I have seen a Theorem: every field contains exactly one prime subfield $K_0$. and next: $K_0$ has to contain 1 and all its multiples: $n \cdot 1 = 1 + \ldots + 1$ Is this because it has to ...
2
votes
2answers
61 views

Is $N_{k\subset K}$ the only *norm* on the field extension $k\subset K$?

In several examples of field extensions the norm function is very useful. For instance, in $\mathbb{Q}\subset\mathbb{Q}(\sqrt{2})$, the norm is $N(x+y\sqrt{2})=x^2-2y^2$. In ...
7
votes
4answers
799 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, a primitive polynomial in $\mathbb Z[x]$ is irreducible in $\mathbb Q[x]$ if and only if it is irreducible in ...
3
votes
1answer
204 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
148 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
66 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
170 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 ...
5
votes
2answers
166 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 ...
3
votes
2answers
231 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
86 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
117 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
269 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
134 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
262 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 ...
3
votes
3answers
628 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
226 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 ...
2
votes
4answers
495 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?
6
votes
1answer
480 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
228 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 $G_f=\mathbb{S}_5$
22
votes
6answers
1k 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
158 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
216 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
100 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
150 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 $X^2-X-a$. ...
7
votes
2answers
554 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
235 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
137 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
186 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 ...
2
votes
2answers
125 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
132 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 ...
7
votes
0answers
438 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
269 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
27 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
308 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
219 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
278 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
84 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
193 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
238 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
260 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
61 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 ...
5
votes
4answers
356 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 ...
7
votes
2answers
132 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
253 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
83 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
182 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 ...
4
votes
2answers
623 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 ...