Fields
http://www.maa.org/taxonomy/term/41402/0
enFields for Which the Principal Axis Theorem Is Valid
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/fields-for-which-the-principal-axis-theorem-is-valid
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>One version of the Principal Axis Theorem asserts that any symmetric matrix with entries in \( \mathcal{R}\) is similar over \(\mathcal{R}\) to a diagonal matrix. The authors find necessary and sufficient conditions for a field \(K\) that make the Principal Axis Theorem valid over \(K\).</em></p>
</div></div></div>Projective Geometry over \(F_1\) and the Gaussian Binomial Coefficients
http://www.maa.org/programs/maa-awards/writing-awards/projective-geometry-over-f1-and-the-gaussian-binomial-coefficients
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal"><b>Year of Award:</b> 2005</p>
<p class="MsoNormal"><b>Publication Information:</b> <i>The American Mathematical Monthly</i>, vol. 111, no. 6, June/July 2004, pp. 487-495.</p></div></div></div>An Algorithm for Multiplication in Modular Arithmetic
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/an-algorithm-for-multiplication-in-modular-arithmetic
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>An algorithm, with examples, for multiplying by \(m\) modulo \( \)n without actually carrying out the multiplications in ordinary arithmetic</em></p>
</div></div></div>Kirkman's Schoolgirls Wearing Hats and Walking Through Fields of Numbers
http://www.maa.org/programs/maa-awards/writing-awards/kirkmans-schoolgirls-wearing-hats-and-walking-through-fields-of-numbers
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Award: </strong>Carl B. Allendoerfer</p>
<p><strong>Year of Award:</strong> 2009</p>
<p><strong>Publication Information: </strong><em>Mathematics Magazine</em>, vol. 82, no. 1, February 2009, pp. 3-15.</p>
<p><strong>Summary:</strong> The historical basis for this interesting article is a problem in recreational mathematics posed by T. P. Kirkman in 1850.</p>
<p>"Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast more than once."</p></div></div></div>Tropical Mathematics
http://www.maa.org/programs/maa-awards/writing-awards/tropical-mathematics
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Award: </strong>Carl B. Allendoerfer</p>
<p><strong>Year of Award:</strong> 2009</p>
<p><strong>Publication Information: </strong><em>Mathematics Magazine</em>, vol. 82, no. 3, June 2009, pp. 163-173.</p>
<p><strong>Summary:</strong> The adjective tropical was chosen by French mathematicians to honor Imre Simon, the Brazilian originator of min-plus algebra, which grew into this field. The basic object of study is the tropical semiring consisting of the real numbers R with a point at infinity under the operations of equals minimum and equals plus.</p></div></div></div>Counting Irreducible Polynomials over Finite Fields Using the Inclusion-Exclusion Principle
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/counting-irreducible-polynomials-over-finite-fields-using-the-inclusion-exclusion-principle
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Using just very basic knowledge of finite fields and the inclusion-exclusion formula, the authors show how one can see the shape of Gauss` formula for the number of irreducible polynomials of a given degree over a finite field .</em></p>
</div></div></div>Matrix Representation of Finite Fields
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrix-representation-of-finite-fields
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>The author uses matrices to represent finite fields in order to make both addition and multiplication obvious.</em></div></div></div>Why Study Equations over Finite Fields?
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/why-study-equations-over-finite-fields
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>The article discusses the reason for studying finite fields.</em></div></div></div>Matrices as Sums of Invertible Matrices
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrices-as-sums-of-invertible-matrices
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The author shows that any square matrix over a field is the sum of two invertible matrices, and that the decomposition is unique only if the matrix is nonzero and of size 2x2 with entries in the field of two elements.</em></p>
</div></div></div>Uncountable Fields Have Proper Uncountable Subfields
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/uncountable-fields-have-proper-uncountable-subfields
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Does an uncountable field have any uncountable proper subfields? This paper shows the answer is "yes, always."</em></p>
</div></div></div>The Existence of Multiplicative Inverse
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-existence-of-multiplicative-inverse
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This capsule discusses a way to show each non-zero element of certain rings have a multiplicative inverse. The approach is to set up a system of linear equations and the solution is the multiplicative inverse. Therefore the non-zero determinant guarantees the existence of the multiplicative inverse.</em></p>
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