minimal polynomial
If a number is algebraic over a field, many polynomials might have it as a root — but among all of them there is one simplest, leanest polynomial that captures it exactly. That is its minimal polynomial: the lowest-degree polynomial with leading coefficient 1 (monic) and coefficients in the base field that the number satisfies. Think of it as the number's algebraic fingerprint.
Precisely, for an element a algebraic over K, its minimal polynomial is the unique monic polynomial m(x) of least degree in K[x] with m(a) = 0. Two key facts: it is irreducible over K (it cannot be factored into smaller-degree polynomials over K), and it divides every other polynomial over K that has a as a root. That divisibility is what makes it “minimal.”
The degree of the minimal polynomial equals the degree of the simple extension K(a)/K — that is, the dimension of K(a) as a vector space over K. So this one polynomial tells you both the irreducible equation a satisfies and exactly how much bigger the field becomes when you adjoin a.
The minimal polynomial of the imaginary unit i over the real numbers is x^2 + 1: it is monic, has i as a root (i^2 + 1 = 0), is irreducible over R, and no degree-1 real polynomial works.
x^2 + 1 is the minimal polynomial of i over R.
The same element can have different minimal polynomials over different base fields. Over the rationals, sqrt(2) has minimal polynomial x^2 - 2; over the reals, where sqrt(2) is already present, its minimal polynomial is just x - sqrt(2).