Let F be a field. Prove that there are infinitely many irreducible monic polynomials

Isa Trevino 2020-10-23 Answered
Let F be a field. Prove that there are infinitely many irreducible monic polynomials
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

SchepperJ
Answered 2020-10-24 Author has 96 answers
To prove the existence of infinitely many monic irreducible polynomials over any given field F.
A polynomial f(x) in F[x] is irreducible if f(x) cannot be factorized as f(x)=g(x)h(x), with g,h in F[x], ie coefficients in F. and deg g and h both greater than 1. In other words, f(x) is irreducbile if it does not be written as a non-trivial product.
If F is an infinite field, there is really nothing to prove: given any aF, x-a is monic and irreducible, as its degree is 1. The set {xa,aF} is therefore an infinite set of monic irreducible polynomials
So the problem is mainly for the case when F is a finite field. In this case, the argument is as shown.
Let F be a finite field, say F=Fq, the finite field with q elements.
Let n be a positive integer.
Then a finite field extension Fqn of degree n.
So, Fqn=Fq(α), where α satisfies a monic irreducible polynomial fn(X)F[x].
Then the set {fn(x):n1} is an infinite set of monic irreducile polynomials over F.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2020-11-20
Prove that in any group, an element and its inverse have the same order.
asked 2021-01-28
Find the Equivalent Polar Equation for a given Equation with Rectangular Coordinates:
rcosθ= 1
asked 2022-04-25

How to solve a cyclic quintic in radicals?
Galois theory tells us that
z111z1=z10+z9+z8+z7+z6+z5+z4+z3+z2+z+1 can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be ζ1,ζ2,,ζ10, following Gauss we can split the problem into solving quintics and quadratics by looking at subgroups of the roots. Since 2 is a generator of the group [2,4,8,5,10,9,7,3,6,1] we can partition into the five subgroups of conjugate pairs [2,9],[4,7],[8,3],[5,6],[10,1].
A0=x1+x2+x3+x4+x5A1=x1+ζx2+ζ2x3+ζ3x4+ζ4x5A2=x1+ζ2x2+ζ4x3+ζx4+ζ3x5A3=x1+ζ3x2+ζx3+ζ4x4+ζ2x5A4=x1+ζ4x2+ζ3x3+ζ2x4+ζx5
Once one has A0,,A4 one easily gets x1,,x5. It's easy to find A0. The point is that τ takes Aj to ζjAj and so takes Aj5 to Aj5. Thus Aj5 can be written down in terms of rationals (if that's your starting field) and powers of ζ. Alas, here is where the algebra becomes difficult. The coefficients of powers of ζ in A15 are complicated. They can be expressed in terms of a root of a "resolvent polynomial" which will have a rational root as the equation is cyclic. Once one has done this, you have A1 as a fifth root of a certain explicit complex number. Then one can express the other Aj in terms of A1. The details are not very pleasant, but Dummit skilfully navigates through the complexities, and produces formulas which are not as complicated as they might be. Alas, I don't have the time nor the energy to provide more details.

asked 2022-04-12
I see the following notation: for a ring A , a prime ideal q ,, and x A.
( q : x ) = ( 1 ) .
The question is: what does ( q : x ) mean? The ideal generated by q , x ? Then why do we need to use a colon?
asked 2021-06-14
i don't get the ideas to solve this question. can anyone help me?
asked 2021-02-25

If U is a set, let G={XXU}. Show that G is an abelian group under the operation defined by XY=(xy)(yx)

asked 2022-01-02
Is there an algorithm for working out the best way (i.e. fewest multiplications) of calculating An in a structure where multiplication is associative?

New questions

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question