micromass said:I wouldn't immediately say that it's obvious, but it's certainly something that you can prove:
Let P(X) be a real polynomial. The fundamental theorem of algebra states that there exists complex number a1,...an such that P(X)=(X-a1)...(X-an). Thus we have factorized the polynomial P over the field C. However, we must factorize it over R.
For this: notice the following fact (try to prove this!): if z is a complex solution of P(X) (thus is P(z)=0), then the complex conjugate [tex]\overline{z}[/tex] is a solution of P(X).
Thus, in our factorization P(X)=(X-a1)...(X-an), if ai is non-real, then one of the a1,...,an must be [tex]\overline{a_i}[/tex]. And if we multiply [tex](X-a_i)(X-\overline{a_i})[/tex], then we get a REAL quadratic polynomial!
Thus P(X) can always be written as a product of linear terms and quadratic terms. This implies that the only possible irreducible polynomials are linear or quadratic.
An irreducible polynomial over the reals is a polynomial that cannot be factored into linear or quadratic polynomials with real coefficients. In other words, it cannot be broken down any further into simpler polynomials.
There is no simple formula or algorithm to determine if a polynomial is irreducible over the reals. However, there are some methods that can be used, such as the rational root theorem, the Eisenstein criterion, or using a computer algebra system.
Yes, a polynomial can be irreducible over the reals but reducible over other fields. For example, the polynomial x2+1 is irreducible over the reals, but it can be factored into (x+i)(x-i) over the complex numbers.
Irreducible polynomials over the reals are important in fields such as algebra, number theory, and cryptography. They also have applications in engineering and physics, particularly in control theory and signal processing.
Yes, an irreducible polynomial over the reals can have complex roots. This is because complex numbers are an extension of the real numbers and can be used to factor polynomials over the reals. In fact, all irreducible polynomials with degree greater than 1 have complex roots.