MHB Is a^2 - 6a + 12 Irreducible Over the Integers?

  • Thread starter Thread starter mathdad
  • Start date Start date
  • Tags Tags
    Integers
Click For Summary
The discussion centers on the polynomial a^2 - 6a + 12, which is stated to be irreducible over the integers, meaning it cannot be factored into polynomials with integer coefficients. The expression is derived from the factorization of 8 + (a - 2)^3. Participants seek clarification on the author's claim and request additional examples of irreducible polynomials. The conversation emphasizes understanding irreducibility in the context of polynomial factorization. Overall, the focus is on the properties of the polynomial and the concept of irreducibility in algebra.
mathdad
Messages
1,280
Reaction score
0
In the textbook, the author showed that 8 + (a - 2)^3 factors out to be a(a^2 - 6a + 12).

The author goes on to say "...the expression a^2 - 6a + 12 is irreducible over the integers."

What does the author means by the statement?
 
Mathematics news on Phys.org
It simply means that the polynomial $a^2-6a+12$ cannot be factored into the product of two polynomials with coefficients that are integers. :D
 
Can you provide me with a few more irreducible examples?
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
View full post »

Similar threads

MHB Factoring x^2 + 64: Is it Possible?
  • · Replies 2 ·
Replies
2
Views
2K
B Calculating the coprime probability of two integers in a different way
  • · Replies 13 ·
Replies
13
Views
2K
MHB Is 7 an irrational number in the set of integers?
  • · Replies 5 ·
Replies
5
Views
3K
MHB Can These Expressions Be Factored Correctly?
  • · Replies 2 ·
Replies
2
Views
2K
I Iterating powers of complex integers along axes of symmetry
  • · Replies 3 ·
Replies
3
Views
1K
MHB Is There an Easier Method to Prove $n^2>n$ for Negative Integers?
  • · Replies 2 ·
Replies
2
Views
2K
I Mutually disjoint sets of all integer powers?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Recreational Number Theory, Unsolved Problem
  • · Replies 1 ·
Replies
1
Views
2K
I Minimize |n-2^x*3^y| over the integer
  • · Replies 5 ·
Replies
5
Views
2K
MHB How to show uniqueness in this statement for integers
  • · Replies 2 ·
Replies
2
Views
1K