Intereting Posts

Adjoint functors as “conceptual inverses”
How prove that $\frac{a_{4n}-1}{a_{2n+1}}$ is integer where $a$ is the Fibonacci sequence
How to deduce the area of sphere in polar coordinates?
Continuous functions on a compact set
How to show $I_p(a,b) = \sum_{j=a}^{a+b-1}{a+b-1 \choose j} p^j(1-p)^{a+b-1-j}$
Let $W$ be a subspace of a vector space $V$ . Show that the following are equivalent.
L : $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear mapping, linear independence of $L$ mapped onto a set of vectors.
Euclidean Geometry Intersection of Circles
Are there necessary and sufficient conditions so that every element in a partially ordered set is either the least element or in the upset of an atom?
Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation
Limit: $\lim_{n\to \infty} \frac{n^5}{3^n}$
norm of integral operator in $C()$
Problem with basic definition of a tangent line.
Showing that a collection of sets is a $\sigma$-algebra: either set or complement is countable
Expectation of maximum of arithmetic means of i.i.d. exponential random variables

I am struggling to find the minimal polynomial for $\displaystyle \frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$.

Does anyone have any suggestions?

Thanks,

Katie.

- Norm computation in number fields
- Intuition and Stumbling blocks in proving the finiteness of WC group
- Problems about consecutive semiprimes
- Algorithm for determining whether two imaginary quadratic numbers are equivalent under a modular transformation
- Archimedean places of a number field
- Intuition in studying splitting and ramification of prime ideals

- Sums of Consecutive Cubes (Trouble Interpreting Question)
- Infinite algebraic extensions of $\mathbb{Q}$
- Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?
- Intuition and Stumbling blocks in proving the finiteness of WC group
- Primes of ramification index 1 with inseparable residue field extension
- On the ring of integers of a compositum of number fields
- What do ideles and adeles look like?
- why does a certain formula in Lang's book on modular forms hold?
- Norm-Euclidean rings?
- Linear independence of fractional powers

One possible procedure that works in general is to first find some polynomial that has this number as its root. E.g. you can start as

$$

\alpha=\frac{\sqrt{2}+5^{1/3}}{\sqrt{3}}\Rightarrow (\sqrt{3}\alpha-\sqrt{2})^3=5,

$$

to get rid of the cube root, then factor out $\sqrt{2}$ and get rid of that, etc. Eventually, you will end up with some polynomial, which might not be irreducible. Decompose it into irreducible factors and check which one of these is satisfied by $\alpha$. If you are clever in deriving the initial polynomial, it will have very few irreducible factors. Also, think in advance what degree you might expect your minimal polynomial to have.

One can compute the minimal polynomial using resultants or Grobner bases. But that is a bit overkill here since it can be done fairly straightforwardly by hand. Namely, let $\rm\ y = \sqrt{2}+\sqrt[3] 5\:.\ $ Then $\rm\: (y-\sqrt 2)^3 = 5\:,\:$ i.e. $\rm\:y^3 + 6\ y – 5 – (3\ y^2 + 2)\ \sqrt{2} = 0\:.\:$ Multiplying that by its conjugate yields $\rm\: y^6 – 6\ y^4 -10\ y^3 + 12\ y^2 -60\ y + 17 = 0\:.\:$ Putting $\rm\ y = \sqrt{3}\ x\:,\:$ then multiplying that by its conjugate yields $\rm\:729\ x^{12} -2916\ x^{10} + 4860\ x^8 – 5670\ x^6 -11340\ x^4 – 9576\ x^2 + 289 = 0\:.$

There is a systematic procedure for finding a polynomial which annihilates an algebraic element of a field extension and which involves no *ad hoc* manipulation.Since it is rarely mentioned in textbooks (except sometimes in exercises), I’ll describe it. It even works for the elements $a\in A$ of a finite-dimensional commutative algebra $A$ over the field $k$.

Consider on the subalgebra $k[a]\subset A$ the endomorphism $\mu_a:k[a]\to k[a]:x\mapsto ax$. It has a characteristic polynomial $\chi (X)=\chi _{\mu_a}(X)=det( XId-\mu_a) \in k[X]$ which, by Cayley-Hamilton’s theorem, annihilates $\mu_a$ and hence also $a$. [Here is an example clarifying the last assertion. If $ \mu_a^3+7\mu_a^4-2Id=0\in End(k[a])$, then, applying the endomorphisms on both sides of that equality to the unit element $1_A$, we get

$a^3+7a^4-2.1_A=0 \in A$]

Now that we have found an annihilating polynomial $\chi (X)$, the minimal polynomial of $a$ is the same as that of $\mu_a$ and can be found, as has already been mentioned, by decomposing $\chi (X)$ into irreducible factors and making finitely many tests to see which divisors of $\chi (X)$ still kill $a$. [Beware that if the algebra $A$ is not a field, the minimal polynomial of $a$ needn’t be irreducible!]

What are the conjugates of this number? Their elementary functions are the coefficients of the minimal polynomial.

*Hint:* The conjugates of $\sqrt 2$ are $\pm \sqrt 2$. The conjugates of $\sqrt 3$ are $\pm \sqrt 3$. The conjugates of $\sqrt[3] 5$ are $\omega \sqrt[3]5$, where $\omega^3=1$. Combine all those and you get all conjugates of the number in question. The minimal polynomial is $\prod (X-\alpha)$, where $\alpha$ runs through the conjugates.

I also used WolframAlpha to get the minimal polynomial

$$289 – 9576x^2 – 11340x^4 – 5670x^6 + 4860x^8 – 2916x^{10} + 729x^{12}$$

Alternately, in Mathematica:

First[RootReduce[(Sqrt[2] + 5^(1/3))/Sqrt[3]]][x] // InputForm

When I’m dealing with lots of algebraics, I always use RootReduce because it’s the most consistent way to eliminate duplicates.

I realize that this question was asked 3 years ago, so I apologize for creating a zombie thread. However, I feel that something major has been missed here, and I want to clarify for anyone Googling and finding this question.

** The question, as asked, is too vague**. What do I mean by this? Well, consider finding the minimal polynomial of $\sqrt{2}$. Of course, if we are in, say, the real numbers $\mathbb{R}$, then the minimal polynomial is simply $f(x) = x – \sqrt{2}$. This is easily verified: the polynomial is of minimal degree, and its coefficients all lie in $\mathbb{R}$, so we are good! However, if I asked you to find the minimal polynomial over $\mathbb{Q}$, then this is a completely different situation, because now our polynomial must have

- $1^n +2^n + \cdots +(p-1)^n \mod p =$?
- Growth of exponential functions vs. Polynomial
- Highest power of a prime $p$ dividing $N!$
- Need help solving complicated integral $\int e^{-x}\cos4x\cos2x\,\mathrm dx$
- Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$
- Alternating sum of a simple product of binomial coefficients: $\sum_{k=0}^{m} (-1)^k \binom m k \binom n k .$
- Calculating probability for forming a triangle
- category definition: class vs set?
- Factorize polynomial over $GF(3)$
- Proving that $\lim_{x\to1^-}\left(\sqrt{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\right)=\Gamma\left(1+\frac1a\right)$
- A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$
- Uncountable product of separable spaces is separable?
- What is $\lim_{n \to \infty} n a_n$?
- Conditional convergence of Riemann's $\zeta$'s series
- Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?