You can try this with pure css, first you need to remove the default behavior of the tag selectusing the property appearance:none.
specified browser
appearance: no;
-webkit-appearance: no; /* chrome and safari */
-moz-appearance: no; /* Mozilla */
-MS-appearance: no; /* Internet explorer */
then you can install background-image
select {
width:100px;
float:left;
appearance:none;
-webkit-appearance:none;
-moz-appearance:none;
-ms-appearance:none;
border:2px solid #000;
background:url('http://www.free-icons-download.net/images/small-down-arrow-icon-15593.png');
background-repeat:no-repeat;
background-size:16px 17px;
background-position:right 0px;
}
<select>
<option>1</option>
<option>2</option>
<option>3</option>
<option>4</option>
</select>
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