To determine the minimal polynomial of a square matrix M over a field K, we look for the monic polynomial of least degree with coefficients in K that evaluates to the zero matrix when applied to M.

Computing the minimal primes of the ring helped us understand its primary decomposition.

She showed that 409 is a minimal prime because none of its digit subsequences form a smaller prime.

In the lecture, the professor illustrated how minimal surfaces arise in soap films spanning wireframes.

When studying algebraic extensions, students often compute minimal polynomials for several elements to understand the field structure.

To determine essential nutrients, the microbiologist tested bacterial growth across different minimal media.

In the phonology seminar, the students compared a near-minimal pair to test whether a proposed phoneme distinction held.

Phonetics students spent the afternoon identifying near-minimal pairs to analyze subtle vowel contrasts.

When reviewing historical medical records, we encountered multiple cases labeled as minimal brain dysfunction.

Startup teams often build minimal viable products to test features quickly and gather user feedback.